Table of Integral Formulas

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.