Download Integrations formula sheet and more Cheat Sheet Calculus in PDF only on Docsity! www.mathportal.org Integration Formulas 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 1 1 n n x x dx C n + = + +∫ 1 lndx x C x = +∫ c dx cx C= +∫ 2 2 x xdx C= +∫ 3 2 3 x x dx C= +∫ 2 1 1 dx C x x = − +∫ 2 3 x x xdx C= +∫ 2 1 arctan 1 dx x C x = + +∫ 2 1 arcsin 1 dx x C x = + − ∫ Integrals of Trigonometric Functions sin cosx dx x C= − +∫ cos sinx dx x C= +∫ tan ln secx dx x C= +∫ sec ln tan secx dx x x C= + +∫ ( )2 1 sin sin cos 2 x dx x x x C= − +∫ ( )2 1 cos sin cos 2 x dx x x x C= + +∫ 2 tan tanx dx x x C= − +∫ 2 sec tanx dx x C= +∫ Integrals of Exponential and Logarithmic Functions ln lnx dx x x x C= − +∫ ( ) 1 1 2 ln ln 1 1 n n n x x x x dx x C n n + + = − + + + ∫ x x e dx e C= +∫ ln x x b b dx C b = +∫ sinh coshx dx x C= +∫ cosh sinhx dx x C= +∫ www.mathportal.org 2. Integrals of Rational Functions Integrals involving ax + b ( ) ( ) ( ) ( ) 1 1 1 n n ax b ax b dx a fo n n r + + + = + ≠ −∫ 1 1 lndx ax b ax b a = + +∫ ( ) ( ) ( )( ) ( ) ( ) 1 2 1 1 2 , 1 2 n na n x b x ax b dx ax b a n n for n n + ≠ − + − + = + + + ≠ −∫ 2 ln x x b dx ax b ax b a a = − + +∫ ( ) ( )2 2 2 1 ln x b dx ax b a ax b aax b = + + ++ ∫ ( ) ( ) ( )( )( ) ( ) 12 1 2 1 , 2 1 n n a n x bx dx ax b a n n for n ax b n − ≠ − − = + − + − − ≠ −∫ ( ) ( ) 22 2 3 1 2 ln 2 ax bx dx b ax b b ax b ax b a + = − + + + + ∫ ( ) 2 2 2 3 1 2 ln x b dx ax b b ax b ax baax b = + − + − ++ ∫ ( ) ( ) 2 2 3 3 2 1 2 ln 2 x b b dx ax b ax baax b ax b = + + − ++ + ∫ ( ) ( ) ( ) ( ) ( ) 3 2 122 3 21 3 2 1 1, 2,3 n n n n ax b b a b b ax bx dx n n fo na r n ax b − − − + + + = − + − − − −+ ≠∫ ( ) 1 1 ln ax b dx x ax b b x + = − +∫ ( )2 2 1 1 ln a ax b dx bx xx ax b b + = − + + ∫ ( ) ( )2 2 2 32 1 1 1 2 ln ax b dx a xb a xb ab x bx ax b + = − + − ++ ∫ Integrals involving ax 2 + bx + c 2 2 1 1 x dx arctg a ax a = + ∫ 2 2 1 ln 1 2 1 ln 2 a x for x a a a x dx x ax a for x a a x a − < + = −− > + ∫
