# Integral calculus formula

Integral calculus is the reverse process of differentiation. So if differentiation of f(x) is g(x) then f(x) is called the integral or anti-derivative of g(x). It is written as $\int g(x)dx = f(x)$

If $\dfrac{d}{dx}f(x) = g(x)$, $\dfrac{d}{dx}f(x) + C = g(x)$ where C is any constant. Thus, if f(x) is the integral of g(x) then f(x)+1 or f(x)+5 or in general f(x)+C is also an integral of g(x). So, based on this fact we write $\int g(x)dx = f(x)+C$ where C is any constant.

INTEGRAL PROPERTIES
1. $\int Cf(x)dx = C \int f(x)dx$
2. $\int {f_1(x) \pm f_2(x)} dx = \int f_1(x) dx \pm \int f_2(x) dx$

BASIC INTEGRAL CALCULUS FORMULA
1. $\int x^n dx = \dfrac{x^{n+1}}{n+1} (n\neq -1)$
2. $\int \frac{1}{x}dx = logx$
3. $\int dx = x$
4. $\int e^x dx = e^x$
5. $\int e^{ax}dx = \dfrac{1}{a}e^{ax}$
6. $\int a^x dx = \dfrac{a^x}{loga} (a>0)$
7. $\int \sin mx dx = -\dfrac{\cos mx}{m}$
8. $\int \cos mx dx = \dfrac{\sin mx}{m}$
9. $\int \sin xdx = -\cos x$
10. $\int \cos xdx = \sin x$
11. $\int \sec ^2xdx = \tan x$
12. $\int cosec^2xdx = -\cot x$
13. $\int \sec x\tan xdx = \sec x$
14. $\int cosec x\cot xdx = -cosec xdx$
15. $\int \sin hxdx = -\cos hx$
16. $\int \cos hxdx = \sin hx$
17. $\int \sec h^2xdx = \tan hx$
18. $\int cosec h^2xdx = -\cot hx$
19. $\int \sec hx\tan hxdx = \sec hx$
20. $\int cosec hx\cot hxdx = -cosec hx$

1. $\int \tan x dx = log (\sec x)$
2. $\int \cot x dx = log (\sin x)$
3. $\int cosecxdx = log (\tan\frac{x}{2}) = log(cosecx-cotx)$
4. $\int secxdx = log tan(\dfrac{\pi}{4}+\dfrac{x}{2}) = log(secx+tanx)$
5. $\int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a} tan^{-1} \dfrac{x}{a}$
6. $\int \dfrac{dx}{x^2 - a^2} = \dfrac{1}{2a} log \dfrac{x-a}{x+a} (x>a)$
7. $\int \dfrac{dx}{a^2 - x^2} = \dfrac{1}{2a} log \dfrac{a+x}{a-x} (x
8. $\int \dfrac{dx}{\sqrt{a^2-x^2}} = sin^{-1}\dfrac{x}{a}$
9. $\int \dfrac{dx}{\sqrt{x^2-a^2}} = log(x+\sqrt{x^2-a^2}) = cosh^{-1}\dfrac{x}{a}$
10. $\int \dfrac{dx}{\sqrt{x^2+a^2}} = log(x+\sqrt{x^2+a^2}) = sinh^{-1}\dfrac{x}{a}$
11. $\int \dfrac{dx}{x\sqrt{x^2-a^2}} = \dfrac{1}{a}sec^{-1}\dfrac{x}{a}$
12. $\int uvdx = u \int vdx - \int (\dfrac{du}{dx}\int vdx)dx$
Here u and v are any two arbitrary functions of x.
13. $\int e^{ax}cosbxdx = \dfrac{e^{ax}(acosbx+bsinbx)}{a^2+b^2}$
14. $\int e^{ax}sinbxdx = \dfrac{e^{ax}(asinbx-bcosbx)}{a^2+b^2}$