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# Integration formulas

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### Integration formulas

1. 1. 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 n ∫ x dx = x n +1 +C n +1 1 ∫ x dx = ln x + C ∫ c dx = cx + C ∫ xdx = x2 +C 2 x3 +C 3 1 1 ∫ x2 dx = − x + C 2 ∫ x dx = ∫ xdx = 1 ∫1+ x ∫ 2 2x x +C 3 dx = arctan x + C 1 1 − x2 dx = arcsin x + C Integrals of Trigonometric Functions ∫ sin x dx = − cos x + C ∫ cos x dx = sin x + C ∫ tan x dx = ln sec x + C ∫ sec x dx = ln tan x + sec x + C 1 ( x − sin x cos x ) + C 2 1 2 ∫ cos x dx = 2 ( x + sin x cos x ) + C ∫ sin 2 ∫ tan ∫ sec x dx = 2 x dx = tan x − x + C 2 x dx = tan x + C Integrals of Exponential and Logarithmic Functions ∫ ln x dx = x ln x − x + C n ∫ x ln x dx = ∫e x x n +1 x n +1 ln x − +C 2 n +1 ( n + 1) dx = e x + C x ∫ b dx = bx +C ln b ∫ sinh x dx = cosh x + C ∫ cosh x dx = sinh x + C
2. 2. www.mathportal.org 2. Integrals of Rational Functions Integrals involving ax + b ( ax + b )n + 1 ∫ ( ax + b ) dx = a ( n + 1) n 1 ( for n ≠ −1) 1 ∫ ax + b dx = a ln ax + b ∫ x ( ax + b ) n a ( n + 1) x − b dx = a x x 2 ( n + 1)( n + 2 ) ( ax + b )n+1 ( for n ≠ −1, n ≠ −2 ) b ∫ ax + b dx = a − a 2 ln ax + b x b 1 ∫ ( ax + b )2 dx = a 2 ( ax + b ) + a 2 ln ax + b a (1 − n ) x − b x ∫ ( ax + b )n dx = a 2 ( n − 1)( n − 2)( ax + b )n−1 ( for n ≠ −1, n ≠ −2 ) 2  x2 1  ( ax + b )  dx = 3 − 2b ( ax + b ) + b 2 ln ax + b  ∫ ax + b  2 a    x2 ∫ ( ax + b )2 x2 ∫ ( ax + b )3 x2 ∫ ( ax + b ) n 1  b2  dx = 3  ax + b − 2b ln ax + b −  ax + b  a    dx = 1  2b b2  ln ax + b + − ax + b 2 ( ax + b )2 a3   dx = 3−n 2− n 1−n 2b ( a + b ) b2 ( ax + b ) 1  ( ax + b ) − + − n−3 n−2 n −1 a3   1 1 ∫ x ( ax + b ) dx = − b ln 1 ax + b x 1 a ∫ x 2 ( ax + b ) dx = − bx + b2 ln 1 ∫ x 2 ( ax + b )2 ax + b x  1 1 2 ax + b dx = − a  2 + 2 − 3 ln  b ( a + xb ) ab x b x  Integrals involving ax2 + bx + c 1 1 x ∫ x 2 + a 2 dx = a arctg a a−x 1  2a ln a + x  ∫ x2 − a 2 dx =  1 x − a  ln  2a x + a  1     for x < a for x > a         ( for n ≠ 1, 2,3)
3. 3. www.mathportal.org 2 2ax + b  arctan  2 4ac − b 2  4ac − b  1 2 2ax + b − b 2 − 4 ac  dx =  ln ∫ ax 2 + bx + c  b 2 − 4ac 2 ax + b + b 2 − 4ac  − 2  2ax + b  x 1 ∫ ax 2 + bx + c dx = 2a ln ax 2 + bx + c − for 4ac − b 2 > 0 for 4ac − b 2 < 0 for 4ac − b 2 = 0 b dx ∫ ax 2 + bx + c 2a m 2an − bm 2ax + b 2 arctan for 4ac − b 2 > 0  ln ax + bx + c + 2 2 2a a 4ac − b 4ac − b  m mx + n 2an − bm 2ax + b  2 2 ∫ ax 2 + bx + c dx =  2a ln ax + bx + c + a b2 − 4ac arctanh b2 − 4ac for 4ac − b < 0  m 2an − bm  ln ax 2 + bx + c − for 4ac − b 2 = 0 a ( 2 ax + b )  2a  ∫ 1 ( ax ∫x 2 + bx + c ) n 1 ( ax 2 + bx + c ) dx = 2ax + b ( n − 1) ( 4ac − b2 )( ax 2 + bx + c ) dx = n−1 + ( 2 n − 3 ) 2a 1 dx 2 ∫ ( n − 1) ( 4ac − b ) ( ax 2 + bx + c )n−1 1 x2 b 1 ln 2 − ∫ 2 dx 2c ax + bx + c 2c ax + bx + c 3. Integrals of Exponential Functions cx ∫ xe dx = ecx c2 ( cx − 1)  x2 2x 2  x 2 ecx dx = ecx  ∫  c − c 2 + c3     ∫x n cx e dx = 1 n cx n n −1 cx x e − ∫ x e dx c c i ∞ cx ( ) ecx dx = ln x + ∑ ∫ x i =1 i ⋅ i ! ∫e cx ln xdx = 1 cx e ln x + Ei ( cx ) c cx ∫ e sin bxdx = cx ∫ e cos bxdx = cx n ∫ e sin xdx = ecx c 2 + b2 ( c sin bx − b cos bx ) ecx c 2 + b2 ( c cos bx + b sin bx ) ecx sin n −1 x 2 c +n 2 ( c sin x − n cos bx ) + n ( n − 1) 2 c +n 2 ∫e cx sin n −2 dx
4. 4. www.mathportal.org 4. Integrals of Logarithmic Functions ∫ ln cxdx = x ln cx − x b ∫ ln(ax + b)dx = x ln(ax + b) − x + a ln(ax + b) 2 2 ∫ ( ln x ) dx = x ( ln x ) − 2 x ln x + 2 x n n n −1 ∫ ( ln cx ) dx = x ( ln cx ) − n∫ ( ln cx ) dx i ∞ ln x ( ) dx = ln ln x + ln x + ∑ ∫ ln x n =2 i ⋅ i ! dx ∫ ( ln x )n =− x ( n − 1)( ln x ) n −1 + 1 dx n − 1 ∫ ( ln x )n −1  ln x 1 x m ln xdx = x m +1  − ∫  m + 1 ( m + 1) 2  ∫ x ( ln x ) m ∫ ( ln x )n x n dx = dx = x m+1 ( ln x ) n m +1 − ( ln x )n+1 ) ( for m ≠ 1) n n −1 m ∫ x ( ln x ) dx m +1 2 ln x n ln x n ( for n ≠ 0 ) ∫ x dx = 2n ln x ln x 1 ∫ xm dx = − ( m − 1) xm−1 − ( m − 1)2 xm−1 ∫ ( ln x )n xm ( for m ≠ 1) ( ln x )n ( ln x )n−1 n dx = − + dx ( m − 1) x m−1 m − 1 ∫ x m dx ∫ x ln x = ln ln x ∞ dx ∫ xn ln x = ln ln x + ∑ ( −1) i =1 dx ∫ x ( ln x )n ∫ ln ( x 2 =− i ( n − 1)i ( ln x )i i ⋅ i! 1 ( for n ≠ 1) ( n − 1)( ln x )n−1 ) ( ) + a 2 dx = x ln x 2 + a 2 − 2 x + 2a tan −1 x ∫ sin ( ln x ) dx = 2 ( sin ( ln x ) − cos ( ln x ) ) x ( for m ≠ 1) ( for n ≠ 1) n +1 (     ( for n ≠ 1) ∫ cos ( ln x ) dx = 2 ( sin ( ln x ) + cos ( ln x ) ) x a ( for m ≠ 1)
5. 5. www.mathportal.org 5. Integrals of Trig. Functions ∫ sin xdx = − cos x ∫ cos xdx = − sin x cos x x 1 − sin 2 x 2 4 x 1 2 ∫ cos xdx = 2 + 4 sin 2 x 1 3 3 ∫ sin xdx = 3 cos x − cos x 1 3 3 ∫ cos xdx = sin x − 3 sin x ∫ sin 2 xdx = dx cos 2 x x ∫ sin x dx = ln tan 2 + cos x ∫ cot 2 xdx = − cot x − x dx ∫ sin x cos x = ln tan x dx x 1 π ∫ sin 2 x cos x = − sin x + ln tan  2 + 4    dx 1 x x ∫ sin x cos2 x = cos x + ln tan 2 x ∫ sin 2 x cos2 x = tan x − cot x ∫ sin x xdx = ln tan 2 dx 1 ∫ sin 2 x dx = − sin x dx π ∫ cos x xdx = ln tan  2 + 4    dx ∫ sin 2 x xdx = − cot x dx ∫ cos2 x xdx = tan x sin( m + n) x sin( m − n) x + 2( m − n) ∫sin mxsin nxdx = − 2( m+ n) cos ( m + n) x cos ( m − n) x − 2( m − n) ∫sin mxcos nxdx = − 2( m + n) sin ( m + n) x sin ( m − n) x + 2( m − n) dx cos x 1 x ∫ sin 3 x = − 2sin 2 x + 2 ln tan 2 ∫ cos mxcos nxdx = 2( m + n) dx sin x 1 x π ∫ cos3 x = 2 cos2 x + 2 ln tan  2 + 4    n ∫ sin x cos xdx = − 1 ∫ sin x cos xdx = − 4 cos 2 x 1 3 2 ∫ sin x cos xdx = 3 sin x 1 2 3 ∫ sin x cos xdx = − 3 cos x x 1 2 2 ∫ sin x cos xdx = 8 − 32 sin 4 x n ∫ sin x cos xdx = ∫ tan xdx = − ln cos x sin x 1 dx = 2 cos x x ∫ cos sin 2 x x π  ∫ cos x dx = ln tan  2 + 4  − sin x   ∫ tan xdx = tan x − x ∫ cot xdx = ln sin x 2 cos n +1 x n +1 sin n +1 x n +1 ∫ arcsin xdx = x arcsin x + 1 − x2 ∫ arccos xdx = x arccos x − 1 − x2 1 ∫ arctan xdx = x arctan x − 2 ln ( x 1 2 ∫ arc cot xdx = x arc cot x + 2 ln ( x 2 ) +1 ) +1 m2 ≠ n2 m2 ≠ n2 m2 ≠ n2