## Mathematics

Published in: Mathematics
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• ### Arman R

• Delhi
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• Teaches: Mathematics, Algebra
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Area,volume and surface formulas,conic section,algebra,numerical and complex,

• 1
1. Common Integrals Indefinite Integral Method of substitution f(g(x))g'(x)dx = f(u)du Integration by parts f(x)g'(x)dx = f(x)g(x) — g(x)f'(x)dx Integrals of Rational and Irrational Functions x 2 2 3 x 3 1 x cxdx = 3 1 dx = arctan x + C 2 —-dx = arcsin x + C 2 1- x Integrals of Trigonometric Functions sin xdx = —cos x + C tan xdx = In sec x + C secxdx = In tan x + sec x + C (x— sin x cos x) + C 1 (x + sin x cos x) + C 2 x dx = tan x — x + C x dx = tan x + C www.mathportal.org Integration Formulas J IPX dx C = CX + C cos xdx = sin x + C Integrals of Exponential and Logarithmic Functions x x In xdx = Inx— b In b x Sinh xdx = cosh x + C cosh xdx = Sinh x + C sin cos tan sec xdx= xdx=
• 2
www.mathportal.org 2. Integrals of Rational Functions Integrals involving ax + b (CIX 4- (CIX +b) n dx = dx = In ax + b CIX 4- b (for n —l) x a ax+b (CIX 4- b) (CIX -k b) n dx = CIX 4- b (CIX 4- l) Y +1 612 b In ax+b a 4- In CIX 4- b a2 (ax+b) a a(l—n)x—b a2 (n —l) (n — 2) (ax + b! - (forn —l, n —2) (forn —l, n —2) I (CIX 4- b) — 21) (CIX 4- b) 4- b2 In ICIX 4- b dx = dx = (CIX 4- b) n dx = x(ax+b) dx = ax + b — 2bln Iax + bl ax+b In Iax + bl + CIX -k b 2 4- b) 3 21) (a +b) 2 n —n (CIX 4- b) CIX -k b CIX -k b + In bx 122 122 (ax+ b) (forn l, 2,3) ax+b 192 (a + xb) ab2x 2 193 Integrals involving ax + bx + c dx = —arctg X 4- Cl dx = x 261 I 261 for x < a Cl 4- X x—a forx > a X -k Cl
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www.mathportal.org 2ax + b for4ac—b2 > 0 4ac — b2 1 for 4ac—b2 < 0 dx = CIX + bX + C for4ac —b2 — 0 x CIX = CIX + bX + C mx+ n dx = CIX + bX + C 1 2 2 4ac — b 2 arctan 2ax + b — In 122 — 4ClC 2ClX + b + 2 2ax + b b — 4ac —4ac dx m m In ax 2a m In ax dx = 2an — bm 2 a 4ac—b 2an — bm 2an — bm a (2ax + b) 2ax + b 2ax + b for 4ac —b2 > 0 arctan 2 4ac — b 2ClX + b for4ac —122 < 0 arctanh for 4ac —122 — 0 (2n-3)2a 1 dx (n —l) (4ac —b2 ax2 + bx + c CIX + bX + C 1 X CIX C 1 In 2 x b 1 n — l) (4ac —b dx 2c CIX -ł-bx+c CIX + bx+c 3. Integrals of Exponential Functions xeCXdx = x2eCXdx xneCXdx = (CX—I) c 2 3 x e dx 00 (CX) dx = In x + ecx In xdx = i=l 1 ecx Inlx + El (CX) (c sin bx — b cos bx) ecx sin bxdx = (c cosbx+ b sin bx) ecx cos bxdx n—l e sin x n (c sin x — n cos bx) + ecx sin n xdx 2 ecx sin n 2 dx
• 4
www.mathportal.org 4. Integrals of Logarithmic Functions Incxdx = xlncx — x In(ax + b)dx = xln(ax + b) — x + In(ax + b) f (In x) 2 x(lnx ) —2xlnx+ 2x f (In cx)n dx = x (In cx —n] (In cx)n- dx 00 (Inx n —1)1 (Inx i.i! = In Inx -k x -k n=2 dx (In x In x x m In xdx = x xm+l (In x) n x m (In x) n dx = (In x) n (In x dx = dx = 2n In x 1 l' dx n—1J (Inx) n 1 1 x m (In x) (forn 1) (forn 0) 1 (In x) n dx dx (forn 1) (for m (for m (form l) (form 1) (In x) n dx = dx = In Iln x dx x n In x dx x(lnx (In x 1 (forn l) 2 —2x + 2a tan Inx + a dx=xln x + a sin (In x)dx = cos (In x)dx = (sin (In x) — cos (In x)) (sin (In x) + cos (In x)) 2
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www.mathportal.org Integrals of Trig. Functions 5. arcsin xdx = x arcsin x + arctan xdx = x arctan x — tanxdx = —In cos x tan xdx = tanx—x sin xdx = —cos x cos xdx = — sin x 1 x sin xdx= sin 2x cos xdx= + — sin 2x sin xdx — cos x— cos x 3 cos xdx= sin x— sin x 3 dx x xdx = In tan sin x 2 dx xdx = In tan + cos x dx xdx = —cot x sin x dx xdx = tan x 1 cos x dx = sin x sin x cos x dx = In tan— + cos x sin x 2 cot 2 xdx = —cot x —x dx = In tan x sin x cos x 1 + In tan + sin dx x cos x dx sin x cos x dx sin xcos x sinmrsinnxdr sin x 2 1 x + In tan 2 cos x = tan x — cot x sin m+nx sin m—n x 4 n) cos m—n x 2(m—n 1712 n2 2(m+n cos m+n)x sinmrcosnxdr 2(m+n cos x dx sin x dx cos x sm m+nx sin x cos x I x + In tan 2 sin x 2 2 sin x + —In tan + 2 cos2x 2 cosmrcosnxdx 2(m+n cosn+l x sin x cos n xdx — sin x sinn xcosxdx 2 sin x cos xdx — sin xcosxdx — sin x cos xdx — 1 cos 2x 4 sin x 3 1 cos arccos xdx — — x arccos x — 1- l— 2 2(m—n 2 x 2 x 3 x sin x cos xdx — 1 32 8 sin x 1 dx = cos x cos x sin x cos x cotxdx = In sin x x sin 4x — sin x arc cot xdx = x arc cot x + —In 2 dx = In tan +

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