{"id":36576,"date":"2022-11-29T12:31:32","date_gmt":"2022-11-29T07:01:32","guid":{"rendered":"https:\/\/tsboardsolutions.com\/?p=36576"},"modified":"2022-11-29T12:31:32","modified_gmt":"2022-11-29T07:01:32","slug":"ts-inter-2nd-year-maths-2b-integration-formulas","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.com\/ts-inter-2nd-year-maths-2b-integration-formulas\/","title":{"rendered":"TS Inter 2nd Year Maths 2B Integration Formulas"},"content":{"rendered":"

Learning these TS Inter 2nd Year Maths 2B Formulas<\/a> Chapter 6 Integration will help students to solve mathematical problems quickly.<\/p>\n

TS Inter 2nd Year Maths 2B Integration Formulas<\/h2>\n

\u2192 A function F(x) Is called an anti derivative or indefinite Integral of a function f(x) If \\(\\frac{d}{d x}\\)[F (x)] = f(x).<\/p>\n

\u2192 Indefinite Integrals:
\nLet f(x) be a function. Then the collection of all its anti derivatives is called the indefinite integral of f(x) and it is denoted by \u222bf(x) dx.
\nThus \\(\\frac{d}{d x}\\) [F(x) + c] = f(x) \u21d4 \u222bf(x)dx = F(x) + c
\nwhere F(x) is the anti derivative and c is the arbitrary constant known as the constant of integration.<\/p>\n

\u2192 Standard forms:
\n\u222bK dx = Kx + c, K is constant<\/p>\n

\u2192 \u222b1 dx = x + c<\/p>\n

\u2192 \u222bxn<\/sup> dx = \\(\\frac{x^{n+1}}{n+1}\\) + c, n \u2260 -1<\/p>\n

\u2192 \u222bx dx = \\(\\frac{x^2}{2}\\) + c<\/p>\n

\u2192 \u222b\u221ax dx = \\(\\frac{2}{3}\\)x\u221ax + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{x}}\\)dx = 2\u221ax + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{x^2}\\)dx = \\(\\frac{-1}{x}\\) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{x}\\)dx = log |x| + c<\/p>\n

\"TS<\/p>\n

\u2192 \u222bex<\/sup>dx = ex<\/sup> + c<\/p>\n

\u2192 \u222bax<\/sup>dx = \\(\\frac{a^x}{\\log a}\\) + c<\/p>\n

\u2192 \u222bsin x dx = – cos x + c<\/p>\n

\u2192 \u222bcos x dx = sin x + c<\/p>\n

\u2192 \u222btan x dx = log|sec x| + c = -log |cos x| + c<\/p>\n

\u2192 \u222bcot x dx = log|sin x| + c = -log|cosec x| + c<\/p>\n

\u2192 \u222bsec x dx = log|sec x + tan x| + c = log|tan\\(\\left(\\frac{\\pi}{4}+\\frac{x}{2}\\right)\\)| + c<\/p>\n

\u2192 \u222bcosec x dx = log|cosec x – cot x| + c = log|tan\\(\\frac{x}{2}\\)| + c<\/p>\n

\u2192 \u222bsec2<\/sup>x dx = tan x + c<\/p>\n

\u2192 \u222bcosec2<\/sup>x dx = -cot x + c<\/p>\n

\u2192 \u222bsec x tan x dx = sec x + c<\/p>\n

\u2192 \u222bcosec x cot x dx = -cosec x + c<\/p>\n

\u2192 \u222bsin hx dx = cos hx + c<\/p>\n

\u2192 \u222bcos hx dx = sin hx + c<\/p>\n

\u2192 \u222btan hx dx = log|cos hx| + c<\/p>\n

\u2192 \u222bcot hx dx = log |sin hx| + c<\/p>\n

\u2192 \u222bsech2<\/sup>x dx = tan hx + c<\/p>\n

\u2192 \u222bcosech2<\/sup>x dx = -cot hx + c<\/p>\n

\u2192 \u222bsec hx tan hx dx = -sec hx + c<\/p>\n

\u2192 \u222bcosec hx cot hx dx = – cosec hx + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{1-x^2}}\\) dx = sin-1<\/sup>x + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{x^2-1}}\\) dx = cosh-1<\/sup>x + c = log(x + \\(\\sqrt{x^2-1}\\)) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{x^2+1}}\\) dx = sin h-1<\/sup>x + c = log(x + \\(\\sqrt{x^2+1}\\)) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{1+x^2}\\)dx = tan-1<\/sup> x + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{1-x^2}\\)dx = \\(\\frac{1}{2}\\)log\\(\\left|\\frac{1+x}{1-x}\\right|\\) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{x^2-1}\\)dx = \\(\\frac{1}{2}\\)log\\(\\left|\\frac{x-1}{x+1}\\right|\\) + c<\/p>\n

\"TS<\/p>\n

\u2192 \u222b\\(\\frac{1}{x \\sqrt{x^2-1}}\\)dx = sec-1<\/sup>x + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{a^2-x^2}}\\)dx = sin-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \\(\\frac{1}{a^2+x^2}\\)dx = \\(\\frac{1}{a}\\)tan-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c = \\(\\frac{-1}{a}\\)cot-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{a^2+x^2}}\\)dx = sinh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c = log(x + \\(\\sqrt{a^2+x^2}\\)) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{x^2-a^2}}\\)dx = cosh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c = log(x + \\(\\sqrt{x^2-a^2}\\)) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{a^2-x^2}\\)dx = \\(\\frac{1}{2 a}\\)log \\(\\left|\\frac{a+x}{a-x}\\right|\\) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{x^2-a^2}\\)dx = \\(\\frac{1}{2 a}\\)log \\(\\left|\\frac{x-a}{x+a}\\right|\\) + c<\/p>\n

\u2192 \u222b\\(\\sqrt{a^2-x^2}\\)dx = \\(\\frac{x}{2} \\sqrt{a^2-x^2}+\\frac{a^2}{2}\\)sin-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \u222b\\(\\sqrt{a^2+x^2}\\)dx = \\(\\frac{x}{2} \\sqrt{a^2+x^2}+\\frac{a^2}{2}\\)sinh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \u222b\\(\\sqrt{x^2-a^2}\\)dx = \\(\\frac{x}{2} \\sqrt{x^2-a^2}-\\frac{a^2}{2}\\)cosh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \\(\\frac{d}{d x}\\)[\u222bf(x)dx] = f(x)<\/p>\n

\u2192 \u222bkf(x) dx = k.\u222bf(x)dx<\/p>\n

\u2192 \u222b[f(x) + g(x)]dx = \u222bf(x)dx + \u222bg(x)dx<\/p>\n

\u2192 \u222b\\(\\frac{f^{\\prime}(x)}{f(x)}\\)dx = log|f(x)| + c<\/p>\n

\u2192 \u222b\\(\\frac{f^{\\prime}(x)}{\\sqrt{f(x)}}\\)dx = 2\\(\\sqrt{f(x)}\\) + c<\/p>\n

\u2192 \u222b[f(x)]n<\/sup>.f'(x)dx = \\(\\frac{f^{n+1}(x)}{n+1}\\) + c, n \u2260 -1<\/p>\n

\u2192 \u222b\\(\\frac{f^{\\prime}(x)}{1+[f(x)]^2}\\)dx = tan-1<\/sup>f(x) + c<\/p>\n

\"TS<\/p>\n

\u2192 Evaluation of Integral of various types by using standard results:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
Type of intrgral<\/td>\nTechnique (or) Substitution<\/td>\n<\/tr>\n
1. \u222b\\(\\frac{1}{a x^2+b x+c}\\)dx (or)
\n\u222b\\(\\frac{1}{\\sqrt{a x^2+b x+c}}\\)dx (or)
\n\u222b\\(\\sqrt{a x^2+b x+c}\\)dx<\/td>\n		Express ax2<\/sup> + bx + c as the sum or difference of two squares. May he reduced to one of the forms like
\n\u222b\\(\\frac{1}{a^2+x^2}\\) dx (or) \u222b\\(\\frac{1}{a^2-x^2}\\) dx (or)
\n\u222b\\(\\frac{1}{x^2-a^2}\\) dx (or) \u222b\\(\\frac{1}{\\sqrt{a^2-x^2}}\\) dx (or)
\n\u222b\\(\\frac{1}{\\sqrt{x^2-a^2}}\\) dx (or) \u222b\\(\\frac{1}{\\sqrt{a^2+x^2}}\\) dx (or)
\n\u222b\\(\\sqrt{a^2+x^2}\\) dx (or) \u222b\\(\\sqrt{a^2-x^2}\\) dx (or)
\n\u222b\\(\\sqrt{x^2-a^2}\\) dx<\/td>\n<\/tr>\n
2. \u222b\\(\\frac{p x+q}{a x^2+b x+c}\\)dx (or)
\n\u222b\\(\\frac{p x+q}{\\sqrt{a x^2+b x+c}}\\)dx (or)
\n\u222b(px + q)\\(\\sqrt{a x^2+b x+c}\\)dx<\/td>\n		Write px + q = A\\(\\frac{d}{d x}\\)(ax2<\/sup> + bx + c) + B where A, B are constants to be determined by equating the coefficients of similar terms on both sides.<\/td>\n<\/tr>\n
3. \u222b\\(\\frac{1}{(p x+q) \\sqrt{a x^2+b x+c}}\\) dx<\/td>\nPut px + q = \\(\\frac{1}{t}\\)<\/td>\n<\/tr>\n
4. \u222b\\(\\frac{1}{\\left(A x^2+B\\right)\\left(C x^2+D\\right)}\\)dx<\/td>\nPut x = \\(\\frac{1}{t}\\)<\/td>\n<\/tr>\n
5. \u222b\\(\\sqrt{(x-\\alpha)(\\beta-x)}\\) dx (or)
\n\u222b\\(\\sqrt{\\frac{x-\\alpha}{\\beta-x}}\\) dx\u00a0 (or)\u222b\\(\\frac{1}{\\sqrt{(x-\\alpha)(\\beta-x)}}\\) dx<\/td>\n		Put x = \u03b1 cos2<\/sup>\u03b8 + \u03b2 sin2<\/sup>\u03b8<\/p>\n

Calculate (x – \u03b1)(\u03b2 -x) and dx<\/td>\n<\/tr>\n

6. \u222b\\(\\frac{1}{a+b \\sin x}\\)dx (or)
\n\u222b\\(\\frac{1}{a+b \\cos x}\\)dx (or)
\n\u222b\\(\\frac{1}{a \\sin x+b \\cos x+c}\\)dx (or)<\/td>\n		Put t = tan\\(\\frac{x}{2}\\)<\/p>\n

sin x = \\(\\frac{2 \\tan \\frac{x}{2}}{1+\\tan ^2 \\frac{x}{2}}\\), cos x = \\(\\frac{1-\\tan ^2 \\frac{x}{2}}{1+\\tan ^2 \\frac{x}{2}}\\)dx = \\(\\frac{2}{1+\\tan ^2 \\frac{x}{2}}\\)dt<\/td>\n<\/tr>\n

7. \u222b\\(\\frac{1}{a+b \\sin ^2 x}\\)dx (or)
\n\u222b\\(\\frac{1}{a+b \\cos ^2 x}\\)dx (or)
\n\u222b\\(\\frac{1}{a \\sin ^2 x+b \\cos ^2 x+c \\cos x \\sin x}\\)dx (or)<\/td>\n		First multiply the numerator and denominator with sec2<\/sup>x and put t = tan x.<\/td>\n<\/tr>\n
8. \u222b\\(\\frac{a \\cos x+b \\sin x}{c \\cos x+d \\sin x}\\)dx<\/td>\nNumerator = A (denominator) + B[\\(\\frac{d}{d x}\\)(denominator)]<\/td>\n<\/tr>\n
9. \u222b\\(\\frac{a \\sin x+b \\cos x+c}{p \\sin x+q \\cos x+r}\\)dx<\/td>\nNumerator = A (denominator) + B\\(\\frac{d}{d x}\\)(denominator) + \u03bb where A, B and \u03bb are constants.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u2192 IntegratIon by parts: If U and V are two functions of x, then
\n\u222bUV dx = U\u222bV dx – \u222b[\\(\\frac{dU}{d x}\\)\u222bVdx]dx<\/p>\n

Extension Rule: \u222bUVdx = UV1<\/sub> – U’V2<\/sub> + U”V3<\/sub> – U”’V4<\/sub> + ……….. + U where U’, U”, U”’ etc., are the successive derivatives of U and V1<\/sub>, V2<\/sub>, V3<\/sub> etc., are successive integrals of V.
\nNote : This rule is very useful if one of the Integrand Is an algebraic function.<\/p>\n

\u2192 Formulae:<\/p>\n