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.