# Differentiation & Integration Formulas With Examples PDF

### Integration Formulas

Integration formulas can be applied for the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, logarithmic and exponential functions.

The integration of functions results in the original functions for which the derivatives were obtained.

These integration formulas are used to find the antiderivative of a function.

If we differentiate a function f in an interval I, then we get a family of functions in I.

If the values of functions are known in I, then we can determine the function f.

This inverse process of differentiation is called integration. Let’s move further and learn about integration formulas used in the integration techniques.

#### What Are Integration Formulas?

The integration formulas have been broadly presented as the following six sets of formulas. Basically, integration is a way of uniting the part to find a whole.

The formulas include basic integration formulas, integration of trigonometric ratios, inverse trigonometric functions, the product of functions, and some advanced set of integration formulas.

Integration is the inverse operation of differentiation.

Thus the basic integration formula is ∫ f'(x).dx = f(x) + C

### List of Integral Formulas PDF

The list of basic integral formulas are

• ∫ 1 dx = x + C
• ∫ a dx = ax+ C
• ∫ xdx = ((xn+1)/(n+1))+C ; n≠1
• ∫ sin x dx = – cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec2x dx = tan x + C
• ∫ csc2x dx = -cot x + C
• ∫ sec x (tan x) dx = sec x + C
• ∫ csc x ( cot x) dx = – csc x + C
• ∫ (1/x) dx = ln |x| + C
• ∫ edx = ex+ C
• ∫ adx = (ax/ln a) + C ; a>0,  a≠1

As we have already gone through integral formulas for exponential functions, logarithmic functions, trigonometric functions, and some basic functions.

Let’s have a look at the additional integration formulas, i.e. the integral formulas for some special functions listed below:

• ∫1(x2–a2)dx=12a.log∣∣(x–a)(x+a)∣∣+C∫1(x2–a2)dx=12a.log|(x–a)(x+a)|+C
• ∫1(a2–x2)dx=12a.log∣∣(a+x)(a–x)∣∣+C∫1(a2–x2)dx=12a.log|(a+x)(a–x)|+C
• ∫1(x2+a2)dx=1atan−1(xa)+C∫1(x2+a2)dx=1atan−1(xa)+C
• ∫1√x2–a2dx=log|x+√x2–a2|+C∫1×2–a2dx=log|x+x2–a2|+C
• ∫1√a2–x2dx=sin−1(xa)+C∫1a2–x2dx=sin−1(xa)+C
• ∫1√x2+a2dx=log|x+√x2+a2|+C