{"id":36775,"date":"2022-12-01T10:35:10","date_gmt":"2022-12-01T05:05:10","guid":{"rendered":"https:\/\/tsboardsolutions.com\/?p=36775"},"modified":"2022-12-03T14:17:20","modified_gmt":"2022-12-03T08:47:20","slug":"maths-2b-integration-important-questions-very-short-answer-type","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.com\/maths-2b-integration-important-questions-very-short-answer-type\/","title":{"rendered":"TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type"},"content":{"rendered":"

Students must practice these Maths 2B Important Questions<\/a> TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type to help strengthen their preparations for exams.<\/p>\n

TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type<\/h2>\n

Question 1.
\nFind \u222b(1 – x)(4 – 3x)(3 + 2x) dx
\nSolution:
\n\"TS<\/p>\n

Question 2.
\nEvaluate \\(\\int \\frac{2 x^3-3 x+5}{2 x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 3.
\nEvaluate \\(\\int \\frac{x^2+3 x-1}{2 x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 4.
\nEvaluate \\(\\int \\frac{(3 x+1)^2}{2 x} d x\\). [(AP) May ’18, ’16]
\nSolution:
\n\"TS<\/p>\n

Question 5.
\nEvaluate \\(\\int\\left(x+\\frac{4}{1+x^2}\\right) d x\\). [(TS) May ’15]
\nSolution:
\n\"TS<\/p>\n

Question 6.
\nEvaluate \\(\\int\\left[\\frac{1}{\\sqrt{1-x^2}}+\\frac{2}{\\sqrt{1+x^2}}\\right] d x\\). [May ’11]
\nSolution:
\n\"TS<\/p>\n

Question 7.
\nEvaluate \\(\\int\\left(x+\\frac{1}{x}\\right)^3 d x\\), x > 0. [May ’12]
\nSolution:
\n\"TS<\/p>\n

Question 8.
\nEvaluate \\(\\int \\frac{x^2+1}{x^4+1} d x\\) on R. [May ’14]
\nSolution:
\n\"TS<\/p>\n

Question 9.
\nEvaluate \\(\\int\\left(\\frac{x^6-1}{1+x^2}\\right) d x\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 10.
\nEvaluate \\(\\int \\frac{\\left(a^x-b^x\\right)^2}{a^x b^x} d x\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 11.
\nEvaluate \\(\\int \\sec ^2 x cosec^2 x d x\\). [(TS) May ’18; Mar. ’16; (AP) May ’17]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 12.
\nEvaluate \\(\\int \\frac{1+\\sin ^2 x}{1+\\cos 2 x} d x\\). [Mar. ’06, May ’95]
\nSolution:
\n\"TS<\/p>\n

Question 13.
\nEvaluate \\(\\int \\frac{1+\\cos ^2 x}{1-\\cos 2 x} d x\\). [Mar. ’19 (TS); ’13]
\nSolution:
\n\"TS<\/p>\n

Question 14.
\nFind \\(\\int \\sqrt{1+\\sin 2 x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 15.
\nEvaluate \\(\\int \\sqrt{1-\\sin 2 x} d x\\). [(TS) May ’17]
\nSolution:
\n\"TS<\/p>\n

Question 16.
\nEvaluate \\(\\int \\sqrt{1-\\cos 2 x} d x\\). [Mar. ’09, May ’06]
\nSolution:
\n\"TS<\/p>\n

Question 17.
\nEvaluate \\(\\int \\frac{d x}{\\sqrt{1+5 x}}\\)
\nSolution:
\n\"TS<\/p>\n

Question 18.
\nFind \\(\\int \\frac{x}{1+x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 19.
\nEvaluate \\(\\int \\frac{e^x}{e^x+1} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 20.
\nEvaluate \\(\\int \\frac{1}{x \\log x[\\log (\\log x)]} d x\\). [Mar. ’19 (TS); ’11]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 21.
\nEvaluate \\(\\int \\frac{1}{x \\log x} d x\\). [May ’99, ’94]
\nSolution:
\n\"TS<\/p>\n

Question 22.
\nEvaluate \\(\\int \\frac{1-\\tan x}{1+\\tan x} d x\\).
\nSolution:
\n\"TS<\/p>\n

Question 23.
\nEvaluate \\(\\int \\frac{\\cos \\sqrt{x}}{\\sqrt{x}} d x\\). [May & Mar. ’98]
\nSolution:
\n\"TS<\/p>\n

Question 24.
\nEvaluate \\(\\int\\left(1-\\frac{1}{x^2}\\right) e^{\\left(x+\\frac{1}{x}\\right)} d x\\). [Mar. ’12]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 25.
\nEvaluate \\(\\int \\frac{e^x(1+x)}{\\cos ^2\\left(x e^x\\right)} d x\\). [(AP) Mar. ’19; May ’17 (TS) Mar. ’17; May ’16]
\nSolution:
\n\"TS<\/p>\n

Question 26.
\nEvaluate \\(\\int \\frac{\\cot (\\log x)}{x} d x\\). [Mar. ’05]
\nSolution:
\n\"TS<\/p>\n

Question 27.
\nEvaluate \\(\\int \\frac{(1+\\log x)^n}{x} d x\\). [(AP) May ’18]
\nSolution:
\n\"TS<\/p>\n

Question 28.
\nEvaluate \\(\\int \\frac{\\log (1+x)}{1+x} d x\\). [(TS) Mar. ’15]
\nSolution:
\n\"TS<\/p>\n

Question 29.
\nEvaluate \\(\\int \\frac{\\sin ^4 x}{\\cos ^6 x} d x\\). [Mar. ’11]
\nSolution:
\n\"TS<\/p>\n

Question 30.
\nEvaluate \\(\\int \\frac{{cosec}^2 x}{(a+b \\cot x)^5} d x\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 31.
\nEvaluate \\(\\int \\frac{1}{\\sqrt{\\sin ^{-1} x} \\sqrt{1-x^2}} d x\\). [(AP) May ’15]
\nSolution:
\nPut sin-1<\/sup>x = t2<\/sup> then
\n\"TS<\/p>\n

Question 32.
\nEvaluate \\(\\int \\frac{\\sin \\left(\\tan ^{-1} x\\right)}{1+x^2} d x\\). [Mar. ’18, ’15 (AP); May ’13]
\nSolution:
\n\"TS<\/p>\n

Question 33.
\nEvaluate \\(\\int \\frac{\\mathrm{e}^{\\tan ^{-1} x}}{1+\\mathrm{x}^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 34.
\nEvaluate \u222bsin mx cos nx dx
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 35.
\nEvaluate \u222bcos mx cos nx dx
\nSolution:
\n\"TS<\/p>\n

Question 36.
\nEvaluate \u222bsin mx sin nx dx
\nSolution:
\n\"TS<\/p>\n

Question 37.
\nEvaluate \u222bcos x cos 3x dx
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 38
\nEvaluate \\(\\int \\frac{d x}{\\sin x+\\sqrt{3} \\cos x}\\). [May ’12]
\nSolution:
\n\"TS<\/p>\n

Question 39.
\nEvaluate \\(\\int \\frac{\\mathbf{x}^2}{\\sqrt{1-x^6}} d \\mathbf{x}\\)
\nSolution:
\n\"TS<\/p>\n

Question 40.
\nEvaluate \\(\\int \\frac{2 x^3}{1+x^8} d x\\). [May ’08]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 41.
\nEvaluate \\(\\int \\frac{x^8}{1+x^{18}} d x\\). [(TS) May ’19; (AP) Mar. ’16]
\nSolution:
\n\"TS<\/p>\n

Question 42.
\nEvaluate \\(\\int \\frac{x^5}{1+x^{12}} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 43.
\nEvaluate \\(\\int \\frac{3 x^2}{1+x^6} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 44.
\nEvaluate \\(\\int \\frac{d x}{(x+5) \\sqrt{x+4}}\\). [May ’09, ’02]
\nSolution:
\n\"TS<\/p>\n

Question 45.
\nFind \\(\\int \\frac{1}{(x+3) \\sqrt{x+2}} d x\\). [May ’14, ’12; Mar. ’14]
\nSolution:
\n\"TS<\/p>\n

Question 46.
\nEvaluate \u222bsec x log(sec x + tan x) dx
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 47.
\nEvaluate \\(\\int \\frac{\\sec ^2 x}{\\sqrt{16+\\tan ^2 x}} d x\\). [May ’08, ’07]
\nSolution:
\n\"TS<\/p>\n

Question 48.
\nEvaluate \\(\\int \\frac{d x}{x^2-81}\\)
\nSolution:
\n\"TS<\/p>\n

Question 49.
\nEvaluate \\(\\int \\frac{3}{\\sqrt{9 x^2-1}} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 50.
\nEvaluate \\(\\int \\frac{d x}{(x+1)(x+2)}\\). [Mar. ’19(AP); Mar. ’14, ’12, May ’11]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 51.
\nEvaluate \\(\\int \\frac{d x}{\\sqrt{x^2+2 x+10}}\\). [May ’06]
\nSolution:
\n\"TS<\/p>\n

Question 52.
\nEvaluate \u222blog x dx. [Mar. ’99, May ’10, ’99]
\nSolution:
\n\"TS<\/p>\n

Question 53.
\nEvaluate \u222bx log x dx. [(AP) Mar. ’20; May ’94]
\nSolution:
\n\"TS<\/p>\n

Question 54.
\nEvaluate \\(\\int \\sin ^{-1} x d x\\). [Mar. ’00, May ’05]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 55.
\nEvaluate \\(\\int x \\tan ^{-1} x d x\\). [Mar. ’05, May ’02]
\nSolution:
\n\"TS<\/p>\n

Question 56.
\nEvaluate \\(\\int e^x \\cos x d x\\). [May ’15 (AP)]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 57.
\nEvaluate \u222bex<\/sup> (sec x + sec x tan x) dx. [(AP) May ’16]
\nSolution:
\n\"TS<\/p>\n

Question 58.
\nEvaluate \u222b(tan x + log sec x) ex<\/sup> dx. [May ’18, ’15 (TS); Mar. ’08, May ’07]
\nSolution:
\n\"TS<\/p>\n

Question 59.
\nEvaluate \\(\\int e^x\\left(\\frac{1+x \\log x}{x}\\right) d x\\). [(TS) May ’19; Mar. ’18, ’15 (AP); Mar. ’13]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 60.
\nEvaluate \\(\\int \\frac{e^x(x+1)}{(x+2)^2} d x\\). [May ’09, ’98]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 61.
\nEvaluate \\(\\int e^x\\left[\\tan ^{-1} x+\\frac{1}{1+x^2}\\right] d x\\). [May ’94]
\nSolution:
\n\"TS<\/p>\n

Question 62.
\nEvaluate \\(\\int e^x\\left(\\tan x+\\sec ^2 x\\right) d x\\). [Mar. ’06, ’00, ’92]
\nSolution:
\n\"TS<\/p>\n

Question 63.
\nEvaluate \u222bex<\/sup> (sin x + cos x) dx. [(AP) Mar. ’17]
\nSolution:
\n\"TS<\/p>\n

Question 64.
\nEvaluate \\(\\int \\frac{1}{1+\\cos x} d x\\). [Mar. ’15 (TS)]
\nSolution:
\n\"TS<\/p>\n

Question 65.
\nEvaluate \\(\\int \\frac{1}{\\cosh x+\\sinh x} d x\\). [(AP) May ’19, ’16 ; Mar. ’17 (TS)]
\nSolution:
\n\"TS<\/p>\n

Question 66.
\nEvaluate \\(\\int\\left(1-x^2\\right)^3 d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 67.
\nEvaluate \\(\\int\\left(\\frac{1}{1-x^2}+\\frac{1}{1+x^2}\\right) d x\\). [(TS) May ’16]
\nSolution:
\n\"TS<\/p>\n

Question 68.
\nEvaluate \\(\\int \\frac{1-\\cos 2 x}{1+\\cos 2 x} d x\\). [May ’02]
\nSolution:
\n\"TS<\/p>\n

Question 69.
\nEvaluate \\(\\int \\sqrt{1+\\cos 2 x} d x\\). [Mar. ’94]
\nSolution:
\n\"TS<\/p>\n

Question 70.
\nEvaluate \\(\\int \\frac{1}{x \\sqrt{x}} d \\mathbf{x}\\). [May ’92]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 71.
\nEvaluate \u222b2x\u221ax dx.
\nSolution:
\n\"TS<\/p>\n

Question 72.
\nEvaluate \u222bx3<\/sup> (4 + x2<\/sup>)2<\/sup> dx. [May ’82]
\nSolution:
\n\"TS<\/p>\n

Question 73.
\nEvaluate \\(\\int \\frac{1-x^4}{1-x} d x\\). [Mar. ’99]
\nSolution:
\n\"TS<\/p>\n

Question 74.
\nEvaluate \\(\\int \\frac{x^4}{x^2+1} d x\\). [May ’98, ’99]
\nSolution:
\n\"TS<\/p>\n

Question 75.
\nEvaluate \\(\\int e^{2 \\log x} d x\\). [May ’99]
\nSolution:
\n\"TS<\/p>\n

Question 76.
\nEvaluate \u222b5x<\/sup> dx. [May ’93]
\nSolution:
\n\\(\\int 5^x d x=\\frac{5^x}{\\log 5}+c\\)<\/p>\n

Question 77.
\nEvaluate \\(\\int \\frac{\\cos ^2 x}{1-\\sin x} d x\\). [Mar. ’01, May ’93]
\nSolution:
\n\"TS<\/p>\n

Question 78.
\nEvaluate \\(\\int \\frac{\\cos x+\\sin x}{\\sqrt{1+\\sin 2 x}} d x\\). [Mar. ’98, May ’92]
\nSolution:
\n\"TS<\/p>\n

Question 79.
\nFind \\(\\int \\frac{x^3}{\\sqrt{x+1}} d x\\). [Mar. ’00]
\nSolution:
\n\"TS<\/p>\n

Question 80.
\nFind \\(\\int \\frac{x^2}{\\sqrt{x+5}} d x\\). [May ’99]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 81.
\nFind \\(\\int \\frac{2 x+3}{\\sqrt{4 x+3}} d x\\). [May ’99]
\nSolution:
\n\"TS<\/p>\n

Question 82.
\nEvaluate \u222bx3<\/sup> sin x4<\/sup> dx. [Mar. ’01]
\nSolution:
\n\"TS<\/p>\n

Question 83.
\nEvaluate \u222bex<\/sup> sin(ex<\/sup>) dx. [(AP) Mar. ’17]
\nSolution:
\nPut ex<\/sup> = t then ex<\/sup> dx = dt
\nNow \u222bex<\/sup> sin(ex<\/sup>) dx = \u222bsin t dt
\n= -cos t + c
\n= -cos(ex<\/sup>) + c<\/p>\n

Question 84.
\nEvaluate \u222b2x sin(x2<\/sup> + 1) dx.
\nSolution:
\nPut x2<\/sup> + 1 = t then 2x dx = dt
\nNow \u222b2x sin(x2<\/sup> + 1) dx = \u222bsin t dt
\n= -cos t + c
\n= -cos(x2<\/sup> + 1) + c<\/p>\n

Question 85.
\nEvaluate \\(\\int \\frac{x}{1+x^4} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 86.
\nEvaluate \\(\\int \\frac{(\\log x)^2}{x} d x\\). [(AP) May ’18]
\nSolution:
\n\"TS<\/p>\n

Question 87.
\nEvaluate \u222bsec(tan x) sec2<\/sup>x dx.
\nSolution:
\nPut tan x = t then sec2<\/sup>x dx = dt
\nNow \u222bsec(tan x) sec2<\/sup>x dx = \u222bsec t dt
\n= log|sec t + tan t| + c
\n= log|sec(tan x) + tan(tan x)| + c<\/p>\n

Question 88.
\nEvaluate \\(\\int \\frac{\\sin (\\log x)}{x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 89.
\nEvaluate \\(\\int \\frac{\\cos (\\log x)}{x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 90.
\nEvaluate \\(\\int \\frac{3 x+7}{3 x^2+14 x-5} d x\\). [Mar. ’00]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 91.
\nEvaluate \\(\\int \\frac{3 \\cos 3 x-2 \\sin 2 x}{\\cos 2 x+\\sin 3 x} d x\\). [May ’93]
\nSolution:
\n\"TS<\/p>\n

Question 92.
\nFind \\(\\int \\frac{6 x}{3 x^2-2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 93.
\nEvaluate \\(\\int \\frac{\\left(\\sin ^{-1} x\\right)^2}{\\sqrt{1-x^2}} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 94.
\nEvaluate \\(\\int \\frac{{cosec}^2 x}{(1+\\cot x)^2} d x\\). [Mar. ’01]
\nSolution:
\n\"TS<\/p>\n

Question 95.
\nEvaluate \u222bcos3<\/sup>x sin x dx. [(TS) Mar. ’18]
\nSolution:
\n\"TS<\/p>\n

Question 96.
\nEvaluate \\(\\int \\sqrt[3]{\\sin x} \\cdot \\cos x d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 97.
\nEvaluate \\(\\int \\sqrt{\\sin x} \\cdot \\cos x d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 98.
\nEvaluate \\(\\int {cosec}^2 x \\cdot \\sqrt{\\cot x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 99.
\nEvaluate \\(\\int \\frac{d x}{4-9 x^2}\\). [May ’98]
\nSolution:
\n\"TS<\/p>\n

Question 100.
\nEvaluate \\(\\int \\frac{1}{\\sqrt{1-4 x^2}} d x\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 101.
\nEvaluate \\(\\int \\frac{d x}{\\sqrt{25+x^2}}\\)
\nSolution:
\n\"TS<\/p>\n

Question 102.
\nEvaluate \\(\\int \\frac{1}{e^x+e^{-x}} d \\mathbf{x}\\)
\nSolution:
\n\"TS<\/p>\n

Question 103.
\nEvaluate \\(\\int \\frac{1}{x^2+6 x+10} d x\\). [Mar. ’98, May ’93]
\nSolution:
\n\"TS<\/p>\n

Question 104.
\nEvaluate \\(\\int \\frac{1}{\\sqrt{x^2-3}} d x\\). [May ’92]
\nSolution:
\n\"TS<\/p>\n

Question 105.
\nEvaluate \\(\\int \\frac{1}{x^2-4} d x\\). [May ’94]
\nSolution:
\n\"TS<\/p>\n

Question 106.
\nEvaluate \\(\\int \\sqrt{x^2+4} d x\\). [May ’93]
\nSolution:
\n\"TS<\/p>\n

Question 107.
\nEvaluate \\(\\int \\sqrt{4 x^2+9} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 108.
\nEvaluate \\(\\int \\sqrt{16-25 x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 109.
\nEvaluate \u222bx ex<\/sup> dx. [Mar. ’99]
\nSolution:
\n\"TS<\/p>\n

Question 110.
\nEvaluate \u222btan-1<\/sup>x dx. [May ’98]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 111.
\nEvaluate \u222bex<\/sup> (cos x – sin x) dx. [May ’99, ’95, Mar. ’99]
\nSolution:
\n\"TS<\/p>\n

Question 112.
\nEvaluate \\(\\int \\frac{x e^x}{(x+1)^2} d x\\). [May ’14, ’98, ’94: Mar. ’05]
\nSolution:
\n\"TS<\/p>\n

Question 113.
\nEvaluate \\(\\int \\frac{1}{\\sqrt{2 x-3 x^2+1}} d x\\). [May ’08]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 114.
\nEvaluate \u222bcot2<\/sup>x dx.
\nSolution:
\n\u222bcot2<\/sup>x dx = \u222b(cosec2<\/sup>x – 1) dx
\n= \u222bcosec2<\/sup>x dx – \u222b1 dx
\n= -cot x – x + c<\/p>\n

Question 115.
\nEvaluate \\(\\int e^{\\log \\left(1+\\tan ^2 x\\right)} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 116.
\nEvaluate \\(\\int \\frac{\\sin ^2 x}{1+\\cos 2 x} d x\\). [(TS) Mar. ’20]
\nSolution:
\n\"TS<\/p>\n

Question 117.
\nEvaluate \u222bsin2<\/sup>x dx.
\nSolution:
\n\"TS<\/p>\n

Question 118.
\nEvaluate \\(\\int \\frac{x}{1+x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 119.
\nEvaluate \\(\\int \\frac{\\sin x}{\\sin (a+x)} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 120.
\nEvaluate \\(\\int 2 x e^{x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 121.
\nEvaluate \\(\\int \\frac{1}{1+\\sin 2 x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 122.
\nEvaluate \u222btan4<\/sup>x sec2<\/sup>x dx.
\nSolution:
\n\"TS<\/p>\n

Question 123.
\nEvaluate \\(\\int \\frac{2 x+3}{\\sqrt{x^2+3 x-4}} d x\\)
\nSolution:
\nPut x2<\/sup> + 3x – 4 = t2<\/sup> then (2x + 3) dx = 2t dt
\n\"TS<\/p>\n

Question 124.
\nEvaluate \u222bsin3<\/sup>x dx.
\nSolution:
\n\"TS<\/p>\n

Question 125.
\nEvaluate \u222bcos3<\/sup>x dx.
\nSolution:
\n\"TS<\/p>\n

Question 126.
\nEvaluate \\(\\int \\frac{d x}{1+e^x}\\)
\nSolution:
\nPut 1 + ex<\/sup> = t then ex<\/sup> = t – 1
\n\u21d2 ex<\/sup> dx = dt
\n\"TS<\/p>\n

Question 127.
\nEvaluate \\(\\int e^x\\left[\\frac{1-\\sin x}{1-\\cos x}\\right] d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 128.
\nEvaluate \u222bx sin2<\/sup>x dx. [Mar. ’02]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 129.
\nEvaluate \\(\\int \\frac{e^x(x+2)}{(x+3)^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 130.
\nEvaluate \\(\\int \\frac{e^{\\mathbf{x}}}{e^x+1} d x\\). [Mar. ’18 (TS)]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 131.
\nEvaluate \\(\\int\\left(x^3-\\cos x+\\frac{4}{\\sqrt{x^2+1}}\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 132.
\nEvaluate \u222b(x3<\/sup> – 2x2<\/sup> + 3) dx.
\nSolution:
\n\"TS<\/p>\n

Question 133.
\nFind \u222b2x7<\/sup> dx.
\nSolution:
\n\"TS<\/p>\n

Question 134.
\nEvaluate \u222b(1 – 2x3<\/sup>) x2<\/sup> dx.
\nSolution:
\n\"TS<\/p>\n

Question 135.
\nEvaluate \\(\\int \\sqrt[3]{2 x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 136.
\nEvaluate \\(\\int\\left(1+\\frac{2}{x}-\\frac{3}{x^2}\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 137.
\nEvaluate \\(\\int \\frac{2 x^3-3 x+5}{2 x^2} d x\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 138.
\nEvaluate \\(\\int\\left(\\frac{3}{\\sqrt{x}}-\\frac{2}{x}+\\frac{1}{3 x^2}\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 139.
\nEvaluate \\(\\int \\frac{1-\\sqrt{x}}{x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 140.
\nEvaluate \\(\\int\\left(\\frac{2 x-1}{3 \\sqrt{x}}\\right)^2 d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 141.
\nEvaluate \\(\\int\\left(e^x-\\frac{1}{x}+\\frac{2}{\\sqrt{x^2-1}}\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 142.
\nEvaluate \\(\\int\\left(\\frac{1}{\\sqrt{x}}+\\frac{2}{\\sqrt{x^2-1}}-\\frac{3}{2 x^2}\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 143.
\nEvaluate \u222b(sec2<\/sup>x – cos x + x2<\/sup>) dx.
\nSolution:
\n\"TS<\/p>\n

Question 144.
\nEvaluate \\(\\int\\left(\\sec x \\tan x+\\frac{3}{x}-4\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 145.
\nEvaluate \\(\\int\\left(\\cosh x+\\frac{1}{\\sqrt{x^2+1}}\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 146.
\nEvaluate \\(\\int\\left(\\sin h x+\\frac{1}{\\left(x^2-1\\right)^{1 \/ 2}}\\right) d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 147.
\nEvaluate \\(\\int(3 x-1)^{1 \/ 2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 148.
\nEvaluate \u222be2x<\/sup> dx
\nSolution:
\n\\(\\int \\mathrm{e}^{2 \\mathrm{x}} \\mathrm{dx}=\\frac{\\mathrm{e}^{2 \\mathrm{x}}}{2}+\\mathrm{c}\\)<\/p>\n

Question 149.
\nEvaluate \\(\\int \\frac{e^{\\log x}}{x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 150.
\nEvaluate \u222bsin 7x dx
\nSolution:
\n\u222bsin 7x dx = \\(\\frac{-\\cos 7 x}{7}+c\\)<\/p>\n

Question 151.
\nEvaluate \u222b(3x2<\/sup> – 4)x dx
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 152.
\n\\(\\int \\frac{1}{7 x+3} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 153.
\nEvaluate \\(\\int \\frac{1}{1+(2 x+1)} d x\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 154.
\nEvaluate \u222bex<\/sup> cot ex<\/sup> dx
\nSolution:
\nPut ex<\/sup> = t \u21d2 ex<\/sup> dx = dt
\n\u2234 \u222bex<\/sup> cot ex<\/sup> dx = \u222bcot t dt
\n= log|sin t| + c
\n= log|sin ex<\/sup>| + c<\/p>\n

Question 155.
\nEvaluate \u222bcot hx dx
\nSolution:
\n\u222bcot hx dx = \\(\\int \\frac{\\cos h x}{\\sin h x} d x\\)
\n= log|sin hx| + c<\/p>\n

Question 156.
\nEvaluate \\(\\int \\frac{\\cos x}{(1+\\sin x)^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 157.
\nEvaluate \u222bcos4<\/sup>x dx
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 158.
\nEvaluate \\(\\int \\frac{x}{\\sqrt{1-x}} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 159.
\nEvaluate \\(\\int x \\sqrt{4 x+3} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 160.
\nEvaluate \\(\\int \\frac{x^2}{\\sqrt{1-x^2}} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 161.
\nEvaluate \\(\\int \\frac{\\sin \\theta}{\\sqrt{2-\\cos ^2 \\theta}} d \\theta\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 162.
\nEvaluate \\(\\int \\frac{3}{\\sqrt{9 x^2-1}} d x\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 163.
\nEvaluate \\(\\int \\frac{1}{1+4 x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 164.
\nEvaluate \\(\\int \\frac{1}{8+2 x^2} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 165.
\nEvaluate \\(\\int\\left(\\sqrt{x}-\\frac{2}{1-x^2}\\right) d x\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 166.
\nEvaluate \\(\\int \\sqrt{9 x^2-25} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 167.
\nEvaluate \\(\\int \\frac{1}{e^x-1} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 168.
\nEvaluate \\(\\int \\frac{1}{1-\\cot x} d x\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 169.
\nEvaluate \\(\\int \\frac{1}{1+\\tan x} d x\\)
\nSolution:
\n\"TS<\/p>\n

Question 170.
\nEvaluate \u222bex<\/sup> (1 + x2<\/sup>) dx
\nSolution:
\n\"TS<\/p>\n

Question 171.
\nEvaluate \u222bcot-1<\/sup>x dx
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 172.
\nEvaluate \u222bsec-1<\/sup>x dx
\nSolution:
\n\"TS<\/p>\n

Question 173.
\nEvaluate \u222bcosec-1<\/sup>x dx
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 174.
\nEvaluate \u222bx2<\/sup> cos x dx
\nSolution:
\n\"TS<\/p>\n

Question 175.
\nEvaluate \u222bx sec2<\/sup>x dx
\nSolution:
\n\"TS<\/p>\n

Question 176.
\nEvaluate \\(\\int \\frac{\\log x}{x^2} d x\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 177.
\nEvaluate \u222b(log x)2<\/sup> dx
\nSolution:
\n\"TS<\/p>\n

Question 178.
\nEvaluate \u222bxn<\/sup> log x dx
\nSolution:
\n\"TS<\/p>\n

Question 179.
\nEvaluate \u222blog(1 + x)2<\/sup> dx
\nSolution:
\n\"TS<\/p>\n

Question 180.
\nEvaluate \u222b\u221ax log x dx. [(TS) Mar. ’16]
\nSolution:
\n\"TS<\/p>\n

Question 181.
\nEvaluate \\(\\int \\mathbf{e}^{\\sqrt{\\mathbf{x}}} d \\mathbf{x}\\). [(AP) May ’19]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 182.
\nEvaluate \u222bcos\u221ax dx. [(TS) May ’17]
\nSolution:
\n\"TS<\/p>\n

Question 183.
\nEvaluate \u222bx cot2<\/sup>x dx
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 184.
\nEvaluate \u222bx2<\/sup> e-3x<\/sup> dx
\nSolution:
\n\"TS<\/p>\n

Question 185.
\nEvaluate \u222bx3<\/sup> eax<\/sup> dx
\nSolution:
\n\"TS<\/p>\n

Question 186.
\nEvaluate \u222bcos (log x) dx
\nSolution:
\n\"TS<\/p>\n

Question 187.
\nIf f(x) is a differentiable function, then prove that \\(\\int \\frac{f^{\\prime}(x)}{f(x)} d x\\) = log|f(x)| + c.
\nSolution:
\nLet f(x) = t
\n\u21d2 f'(x) dx = dt
\n\u2234 \\(\\int \\frac{f^{\\prime}(x)}{f(x)} d x=\\int \\frac{1}{t} d t\\)
\n= log|t| + c
\n= log|f(x)| + c<\/p>\n

Question 188.
\nIf f(x) is a differentiable function and n \u2260 -1, then prove that \\(\\int[f(x)]_{f^{\\prime}(x)}^n \\cdot d x=\\frac{[f(x)]^{n+1}}{n+1}+c\\)
\nSolution:
\nPut f(x) = t
\n\u21d2 f'(x) dx = dt
\n\"TS<\/p>\n

Question 189.
\nProve that \u222btan x dx = log|sec x| + c.
\nSolution:
\n\"TS<\/p>\n

Question 190.
\nProve that \u222bcot x dx = log|sin x| + c.
\nSolution:
\n\"TS<\/p>\n

Question 191.
\nProve that \u222bsec x dx = log|sec x + tan x| + c = \\(\\log \\left|\\tan \\left(\\frac{\\pi}{4}+\\frac{x}{2}\\right)\\right|\\) + c
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 192.
\nProve that \u222bcosec x dx = log|cosec x – cot x| + c = \\(\\log \\left|\\tan \\frac{x}{2}\\right|\\) + c
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 193.
\nProve that \u222bex<\/sup> [f(x) + f'(x)] dx = ex<\/sup> f(x) + c
\nSolution:
\n\"TS<\/p>\n

Question 194.
\nIf \\(I_n=\\int x^n \\cdot e^{-x} d x\\), then prove that \\(I_n=-x^n \\cdot e^{-x}+n I_{n-1}\\).
\nSolution:
\n\"TS<\/p>\n","protected":false},"excerpt":{"rendered":"

Students must practice these Maths 2B Important Questions TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type to help strengthen their preparations for exams. TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type Question 1. Find \u222b(1 – x)(4 – 3x)(3 + 2x) dx Solution: Question 2. … Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[26],"tags":[],"yoast_head":"\nTS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type - TS Board Solutions<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/tsboardsolutions.com\/maths-2b-integration-important-questions-very-short-answer-type\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type - TS Board Solutions\" \/>\n<meta property=\"og:description\" content=\"Students must practice these Maths 2B Important Questions TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type to help strengthen their preparations for exams. 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TS Inter Second Year Maths 2B Integration Important Questions Very Short Answer Type Question 1. Find \u222b(1 – x)(4 – 3x)(3 + 2x) dx Solution: Question 2. ... 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