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

ADVANCE INTEGRAL CALCULUS FORMULA
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<a)
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}