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Indefinite Integral Table
Note: u, v, and w are functions of x. a, c, n, are constants. All trigonometric functions use radians. A constant must be added 4. to the result of every integration. General and Basic Integrals 1. ∫ af ( x ) dx = a ∫ f ( x ) dx 2. ∫ ( u ± v ) dx = ∫ u dx ± ∫ v dx 3. ∫ u dv = uv – ∫ v du g′ ( x ) 5. ∫ ------------ dx = ln g ( x ) g(x) { [ g ( x) ]r + 1 } ⁄ ( r + 1 ) r≠ 1 6. ∫ [ g ( x ) ] r g′ ( x ) dx =  ln g ( x ) r = 1  x n + 17. ∫ x n dx = ----------n+1 dx 8. ∫ ---- = ∫ x –1 dx = x dx 9. ∫ ---- = ∫ x –n dx = xn SIN 1 = – -- cos ax a 1 1 1 1 sin 2ax 2 2 3. ∫ sin x dx = -- x – -- sin 2x 4. ∫ sin ax dx = -- x – -- ---------------2 4 2 4 a 3 1 3 4 --- sin x cos x – 3 sin x cos x + -- x 5. ∫ sin x dx = – 8 8 4 n–1 1 n–1 n n–2 6. ∫ sin x dx = – -- sin x cos x +----------- ∫ sin x dx n n 1. ( n ≠ –1 ) ln x (x ≠ 0) 5.

www.et.byu.edu/~jww8
n–2 1 n n–2 n–2 ∫ csc x dx = – ----------- csc x cot x + ----------- ∫ csc x dx n–1 n–1

© 2002 by Jon Wittwer
1 2 2c. ∫ ----------------------------- dx = – -----------------ax 2 + bx + c 2ax + b Integrals Involving a 2 ± b 2 x 2 –1 1 1 bx 1. ∫ --------------------2 dx = ----- tan ----a2 + b2x ab a b 2 – 4ac = 0∫ csc x dx = x csc x + ln x + x 2 – 1
–1 –1

2 ∫ a + bx dx = ------ ( a + bx ) 3 / 2 3b 2 2. ∫ x a + bx dx = ----------2 ( 3bx – 2a ) ( a + bx ) 3 / 2 15b 1. 2x n ( a + bx ) 3 / 2 2an 3. ∫ x n a + bx dx = --------------------------------- – ----------------------- ∫ x n – 1 a + bx dx b ( 2n + 3 ) b ( 2n + 3 ) x 2 4. ∫ ------------------ dx = -------2 ( bx – 2a ) a + bx 3b a + bx xn 2x n a +bx 2an - x n – 1 5. ∫ ------------------ dx = --------------------------- – ----------------------- ∫ ------------------ dx b ( 2n + 1 ) b ( 2n + 1 ) a + bx a + bx  1 a + bx – a  ------ ln -------------------------------- for ( a > 0 )  a a + bx + a 1 6. ∫ ---------------------- dx =  –1 x a + bx 2  --------a + bx - tan -------------- for ( a < 0 )  –a  –a b ( 2n – 3 ) 1 1 a + bx 7. ∫------------------------ dx = – ------------------------------ – ----------------------- ∫ ----------------------------- dx a ( n – 1 )x n – 1 2a ( n – 1 ) x n – 1 a + bx x n a + bx 1 a + bx 8. ∫ ------------------ dx = 2 a + bx + a ∫ ---------------------- dx x x a + bx a + bx ( a + bx ) 3 / 2 b ( 2n – 5 ) a + bx 9. ∫ ------------------ dx = – ------------------------------ – ---------------------- ∫------------------ dx xn a ( n – 1 )x n – 1 2a ( n – 1 ) x n – 1 1 2 10. ∫ -------------------------------------- dx = --------- tanh ax + b cx + d ac Integrals Involving a – x
2 2 2 –1

Combined Trig Functions 2 1. ∫ sin x cos x dx = ( sin x ) ⁄ 2 cos ( a – b ) x cos ( a + b ) x 2. ∫ sin ax cos bx dx = – ---------------------------- – ---------------------------2(a – b) 2(a + b) 3.

( a > 0,b > 0 )

–1 1 a + bx 11 bx 2. ∫ -------------------- dx = ----- tanh ----- = --------- ln  --------------  2ab  a – bx  ab a 2 – b 2 x2 a

∫ sec x tan x dx

= sec x
m+1

4.

∫ csc x cot x dx
n–1 n+1

= – csc x

for ( a > 0, b > 0 ) 3.

5a.

sin x cos x n – 1 m n m n–2 ∫ sin x cos x dx = -------------------------------------- + ------------- ∫ sin x cos x dx m+n m+n
m

∫a2 x bx b2x2 a 2 + b 2 x 2 dx = -- a 2 + b 2 x 2 + ----- ln  ----- + 1 + ---------  2 2b  a a2 

∫ sin

sin x cos x m – 1 n m–2 n x cos x dx = – ------------------------------------- + ------------ ∫ sin x cos x dx m+n m+n

m–1

–1 1 1 1 b 6. ∫ ----------------------------------------- dx = ----------------------- ln tan --  cx + tan -- 2 a sin cx + b cos cx a c a2 + b2

x1 –n ----------- ( n ≠ 1 ) 1–n

∫ sin x dx

= – cos x

2.

∫ sin ax dx

7. ∫ x sin x dx = sin x – x cos x sin ax x cos ax 8. ∫ x sin ax dx = ------------- – ----------------2 a a 9. ∫ x 2 sin x dx = – x 2 cos x + 2x sin x + 2 cos x 10. ∫ x sin x dx = – x cos x + n ∫ x
n n n–1

Integrals Involving e x e ax 1 F(u) ∫ F ( e a x ) dx = -- ∫ ----------- du, u = e a x a u 1 1. ∫ e x dx = e...
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