Integration formula topic wise. Integration Formulas List

**integration formula calculus formula **is a most using** formula** for** calculus in mathematic** Actually,** Integration is the reverse process of Differentiation**. Therefore, it is essential to know the formula as well as the differential to solve the **questions of integration**. **Integration Formulas** is the process with the help of which any type of integration related questions can be solved easily.

Calculus is the biggest topic of **Maths in class 12th**. It covers about 44% part of **Mathematics **which includes topics like Differential, Integration, Santana, Derivative Equation etc. Among these, the biggest topic is Integration, which is very interesting. It is impossible to study it or solve the question without **Integration Formulas**.

Here all the integration formulas from Basic to Advance are displayed which help the most in solving the questions.

## Table of Contents

## What is Integration?

In general, integration is the inverse process of differentiation, also known as inverse differentiation. It is actually a process of working opposite to the differential function which is called integration.

Thus, finding the integral of a function f (x) means finding a function f (x) whose derivative is f (x). Some special facts regarding this are as follows.

f'(x).dx = f(x) + C

**Integral:** The function whose integral is to be found is called integral.

**Integration:** The method of finding the integral of a function is called Integration.

**Integral:** The function whose derivative is integral is called the integral of the integral.

∫ 1 dx | x + C |

∫ a dx | ax + C |

∫ x^{n }dx | ((x^{n+1})/(n+1)) + C |

∫ sin x dx | – cos x + C |

∫ cos x dx | sin x + C |

∫ sec^{2}x dx | tan x + C |

∫ cosec^{2}x dx | – cot x + C |

∫ sec x (tan x) dx | sec x + C |

∫ cosec x ( cot x) dx | – cosec x + C |

∫ (1/x) dx | log |x| + C |

∫ e^{x }dx | e^{x}+ C |

∫ a^{x }dx | (a^{x} / log a) + C |

∫ tan x dx | log | sec x | + C |

∫ cot x dx | log | sin x | + C |

∫ sec x dx | log | sec x + tan x | + C |

∫ cosec x dx | log | cosec x – cot x | + C |

∫ 1 / √ ( 1 – x^{2} ) dx | sin ^{–} ^{1} x + C |

∫ 1 / √ ( 1 – x^{2} ) dx | cos ^{–} ^{1} x + C |

∫ 1 / √ ( 1 + x^{2} ) dx | tan ^{–} ^{1} x + C |

∫ 1 / √ ( 1 + x^{2} ) dx | cot ^{–} ^{1} x + C |

## Sectional Method Formula | Method of Parts Formula

Generally, the correct choice of u and v is to be done to find the integration by the block method. Therefore, a formula works in relation to this, which is represented by the name of ILATE. Its meaning is defined as follows.

**I**= Inverse Trigonometry Function**L**= Logarithm Function**A**= Algebraic Function**T**= Trigonometry Function**E**= Exponential Function

## Integration Formula List | Integration Formulas in English

Integration is the most important topic in Calculus, so, it is our responsibility to get specific information about it so that it is easy to solve the question. Here the list of all the formulas is provided in a systematic manner which is essential for class 12.

- ∫1 dx = x + C
- ∫ a dx = ax+ C
- ∫ (1/x) dx = ln |x| + C
- ∫ ex dx = ex+ C
- ∫ 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
- ∫cosec2x.dx = -cotx + C
- ∫secx.tanx.dx = secx + C
- ∫cosecx.cotx.dx = -cosecx + C
- ∫tanx.dx =log|secx| + C
- ∫cotx.dx = log|sinx| + C
- ∫secx.dx = log|secx + tanx| + C
- ∫cosecx.dx = log|cosecx – cotx| + C
- ∫ ax dx = (ax/ln a) + C ; a>0, a≠1

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## Generally, integration is classified into four major parts, therefore, we will study Integration Formulas according to the classification.

- Method of Transformation
- Method of Substitution
- Method of Parts
- Method of Partial Fractions

## Basic integration formula

- ∫ xn.dx = x(n + 1)/(n + 1)+ C
- ∫1.dx = x + C
- ∫ ex.dx = ex + C
- ∫1/x.dx = log|x| + C
- ∫ ax.dx = ax /loga+ C
- ∫ ex[f(x) + f'(x)].dx = ex.f(x) + C

## Integration Formulas of Inverse Trigonometric functions:

- ∫1/√(1 – x
^{2}).dx = sin^{-1}x + C - ∫ /1(1 – x
^{2}).dx = -cos^{-1}x + C - ∫1/(1 + x
^{2}).dx = tan^{-1}x + C - ∫ 1/(1 +x
^{2}).dx = -cot^{-1}x + C - ∫ 1/x√(x
^{2}– 1).dx = sec^{-1}x + C - ∫ 1/x√(x
^{2}– 1).dx = -cosec^{-1}x + C

## Advanced Integration Formulas

- ∫1/(x
^{2}– a^{2}).dx = 1/2a.log|(x – a)(x + a| + C - ∫ 1/(a
^{2}– x^{2}).dx =1/2a.log|(a + x)(a – x)| + C - ∫1/(x
^{2}+ a^{2}).dx = 1/a.tan-1x/a + C - ∫1/√(x
^{2}– a^{2})dx = log|x +√(x^{2}– a^{2})| + C - ∫ √(x
^{2}– a^{2}).dx =1/2.x.√(x^{2}– a^{2})-a^{2}/2 log|x + √(x2 – a2)| + C - ∫1/√(a
^{2}– x^{2}).dx = sin-1 x/a + C - ∫√(a
^{2}– x^{2}).dx = 1/2.x.√(a^{2}– x^{2}).dx + a2/2.sin-1 x/a + C - ∫1/√(x
^{2}+ a^{2}).dx = log|x + √(x^{2}+ a^{2})| + C - ∫ √(x
^{2}+ a^{2}).dx =1/2.x.√(x^{2}+ a^{2})+ a2/2 . log|x + √(x^{2}+ a^{2})| + C

sin^{2} x | ( 1 – cos 2x ) / 2 |

cos^{2} x | ( 1 + cos 2x ) / 2 |

sin^{3} x | ( 3 sin x – sin 3x ) / 4 |

cos^{3} x | ( 3 cos x + cos 3x ) / 4 |

tan^{2} x | sec^{2} x – 1 |

sin^{2} x + cos^{2} x | 1 |

tan^{2} x | cosec^{2} x – 1 |

2sin A . sin B | cos(A – B) + cos(A + B) |

2sin A . cos B | sin(A + B) + sin(A – B) |

2cos A . sin B | sin(A + B) – sin(A – B) |

2cos A . cos B | cos(A + B) + cos(A – B) |

Sin 3x | 3sin x – 4sin^{3} x |

Cos 3x | 4cos^{3} x – 3cos x |

Tan 3x | ( 3tan x – tan^{3} x ) / ( 1 – 3tan^{2} x ) |

sin 2x | 2sin x • cos x = 2tan x / (1+tan^{2} x ) |

cos 2x | cos^{2} x – sin^{2} x = (1- tan^{2} x ) / ( 1+tan^{2} x ) |

cos 2x | 2cos^{2} x −1 = 1 – 2sin^{2} x |

tan 2x | ( 2tan x ) / (1−tan^{2} x ) |

sec 2x | sec^{2 }x / (2-sec^{2} x ) |

Cosec 2x | (sec x . Cosec x ) / 2 |