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Table of Integrals

A table of indefinite integrals of functions is presented below.

In what follows, c is a constant of integration and can take any constant value.

1 - Integrals of Elementary Functions.


1.1   
dx = x + c

1.2   
k dx = k x + c , where k is a constant.

1.3   
x n dx = x n + 1 / (n + 1) + c

1.4   
(1 / x) dx = ln |x| + c

2 - Integrals of Elementary Trigonometric Functions : sin x, cos x, tan x, cot x, sec x and csc x.


2.1   
sin x dx = -cos x + c

2.2   
cos x dx = sin x + c

2.3   
tan x dx = ln |sec x| + c

2.4   
cot x dx = ln |sin x| + c

2.5   
sec x dx = ln |sec x + tan x| + c

2.6   
csc x dx = ln |csc x - cot x| + c

3 - Integrals Involving More Than One Trigonometric Function.


3.1   
sec x tan x dx = sec x + c

3.2   
csc x cot x dx = -csc x + c

3.3   
sin mx sin nx dx =

-sin [(m + n)x] / 2(m + n) + sin [(m - n)x] / 2(m - n) + c

with m not equal to n.

3.4   
cos mx cos nx dx =

sin [(m + n)x] / 2(m + n) + sin [(m - n)x] / 2(m - n) + c

with m not equal to n.

3.5   
sin mx cos nx dx =

-cos [(m + n)x] / 2(m + n) - cos [(m - n)x] / 2(m - n) + c

with m not equal to n.

4 - Integrals Involving Exponential and Logarithmic Functions.


4.1   
e x dx = e x + c

4.2   
a x dx = a x / ln a + c

4.3   
ln x dx = x ln x - x + c

5 - Integrals of Inverse Trigonometric functions: arcsin x, arccos x, arctan x, arccot x, arcsec x and arccsc x.


5.1   
arcsin x dx = x arcsin x + sqrt (1 - x 2) + c

5.2   
arccos x dx = x arccos x - sqrt (1 - x 2) + c

5.3   
arctan x dx = x arctan x - ln [sqrt (1 + x 2)] + c

5.4   
arccot x dx = x arccot x + ln sqrt (1 + x 2) + c

5.5   
arcsec x dx = x arcsec x - ln [x + sqrt (x 2 - 1)] + c

5.6   
arccsc x dx = x arccsc x + ln [x + sqrt (x 2 - 1)] + c

6 - Integrals Involving Exponential and Sine and Cosine Functions.


6.1   
e ax sin bx dx = (e ax / (a 2 + b 2) (a*sin bx - b*cos bx) + c

6.2   
e ax cos bx dx = (e ax / (a 2 + b 2) (b*sin bx + a*cos bx) + c

7 - Integrals Involving Hyperbolic Functions: sinh x, cosh x, tanh x, coth x, sech x, csch x.


7.1   
sinh x dx = cosh x + c

7.2   
cosh x dx = sinh x + c

7.3   
sech x tanh x dx = -sech x + c

7.4   
csch x coth x dx = -csch x + c

7.5   
sech 2 x dx = tanh x + c

7.6   
csch 2 x dx = -coth x + c



More references on
integrals and their applications in calculus.



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Updated: 2 April 2013

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