Complete 100+ Integration Formulas Basic to Advanced

Integration Formulas are provided here . These formulas will help the student to quickly revise their subject of Integration. These are helpful in exams

Here are the complete 100+ Integration formulas. This post contains all the integration formulas from basic to Advance.

Integrals / Anti Derivative

A function ϕ(x) is said to be anti derivative of a function if ϕ'(x) = f(x).
\( \frac{d}{dx}{\varphi(x)}\ =\ f(x) \)

Indefinite Integrals

If a f(x) is a function , then the family of all its antiderivatives is called Indefinite Integrals. The general form of an Indefinite Integrals is provided below :-
\(\int{f(x).dx} \)
It is called indefinite integral because it is not unique. The value of definite integrals is unique. An indefinite integral does not have any limit to apply. But a definite integral has upper limits and lower limits.
Indefinite Integrals – \(\int{f\left(x\right).dx}=\varphi\left(x\right)+C \)
Definite Integrals – \(\int_{a}^{b}{{f(x)}.dx\ }=\ {\varphi(x)}_a^b\ =\ \varphi(b)\ -\ \varphi(a)\ \)

Standard Integration Formulas

These are the basic integration formulas that are used to solve the problems of integration functions.

1. \(\int x^n.dx\ =\ \frac{x^{n+1}}{n+1}\ +\ C \)

2. \(\int1.dx\ =\ x\ +C \)

3. \(\int\frac{1}{x}.dx\ =\ \log{x}\ +C \)

4. \(\int\cos{x}\ .\ dx\ =\ \sin{x}\ +\ C \)

5. \(\int\sin{x}\ .dx\ =\ -\cos{x}\ +\ C \)

6. \(\int{\sec^2x\ }.dx\ =\ tan\ x\ +C\ \)

7. \(\int{{\rm cosec}^2x}\ .\ dx\ =\ -\ cot\ x\ +\ C \)

8. \(\int{sec\ x\ tan\ x}\ .\ dx\ =\ sec\ x\ +\ C \)

9. \(\int{cosec\ x\ cot\ x\ }.dx\ =\ -\ cosec\ x\ +\ C \)

10. \(\int e^x\ .dx\ =\ e^x\ +\ C \)

11. \(\int a^x.dx\ =\ \frac{a^x}{\log{a}}\ +\ C \)

12. \(\int\frac{1}{\sqrt{1\ -\ x^2}}\ .dx\ =\ \sin^{-1}{\frac{x}{1}}\ +C \)

13. \(\int\frac{1}{\sqrt{a^2\ -\ x^2}}\ .dx\ =\ \sin^{-1}{\frac{x}{a}}\ +C \)

14. \(\int{\frac{1}{1+x^2}\ }.dx\ =\ \tan^{-1}{x}\ +\ C \)

15. \(\int{\frac{1}{a^2\ +\ x^2}\ }.dx\ =\ \frac{1}{a}\ \tan^{-1}{\frac{x}{a}}\ +\ C \)

16. \( \int{\frac{1}{x\sqrt{x^2-1}}\ .dx}=\ \sec^{-1}{x}\ +\ C\)

17. \(\int{-\frac{1}{x\sqrt{x^{2\ }-\ 1}}.dx=}{\rm cosec}^{-1}{x}+\ C \)

18. \( \int{-\frac{1}{a^2\ +\ x^2}.dx\ =\ \frac{1}{a}\ }\cot^{-1}{\frac{x}{a}}\ +\ C \)

19. \(\int{\frac{1}{x\sqrt{x^2\ -\ a^2}}\ .dx}\ =\ \frac{1}{a}\ \sec^{-1}{\frac{x}{a}}\ +\ C \)

20. \(\int{-\frac{1}{x\sqrt{x^2\ -\ a^2}}\ .dx}\ =\ \frac{1}{a}\ co\sec^{-1}{\frac{x}{a}}\ +\ C \)

Integration by Substitution

The method of evaluating integrals of a function by substituting a suitable function is known as integration by substitution . Here are some of the integrals which are used in the method of Integration by substitution. In these fundamental integrals, x is replace by ” ax + b ” .

Integration Formulas For Substitution method

1. \(\int{\left(ax+b\right)^n\ .dx\ }=\ \frac{\left(ax+b\right)^{n+1}}{a\left(n+1\right)}\ +C\ \)

2. \(\int{\frac{1}{ax\ +\ b\ }\ .dx}\ =\frac{1}{a}\ \log{\left(ax+b\right)}\ +C\ \)

3. \(\int{e^{\left(ax+b\right)}.dx}\ =\ \frac{1}{a}e^{\left(ax+b\right)}\ +\ C \)

4. \(\int{a^{\left(ax+b\right)}.dx}\ =\ \frac{1}{b}\times\frac{a^{\left(bx+c\right)}}{\log{a}}\ +\ C \)

5. \(\int\sin{\left(ax+b\right).dx} \)
\( =-\frac{1}{a}cos\left(ax+b\right)+C \)

6. \(\int{cos\left(ax+b\right).dx} \)
\( =\frac{1}{a}sin\left(ax+b\right)+C \)

7. \(\int{\sec^2\left(ax+b\right).dx} \)
\( =\frac{1}{b}tan\left(ax+b\right)+C \)

8. \(\int{{\rm cosec}^2\left(ax+b\right).dx} \)
\( =-\frac{1}{a}cot\left(ax+b\right)+C \)

9. \(\int{sec\left(ax+b\right)tan\left(ax+b\right).dx} \)
\( =\frac{1}{a}sec\left(ax+b\right)+C \)

10. \(\int{cosec\left(ax+b\right)cot\left(ax+b\right).dx} \)
\( =-\frac{1}{a}cosec\left(ax+b\right)+C \)

11. \(\int{tan\left(ax+b\right).dx}\ \)
\( =\ -\frac{1}{a}log\left|cos\left(ax+b\right)\right|\ +C\ \)

12. \(\int{cot\left(ax+b\right).dx} \)
\(=\frac{1}{a}log\left|sin\left(ax+b\right)\right|\ +\ C\ \)

13. \( \int{tan\ x.dx}\ \)
\( =\ -\ log\left|cos\ x\right|\ +\ C\ \ \)
\( =\ log\left|sec\ x\right|\ +\ C \)

14. \(\int{cot\ x\ .\ dx}\ =\ log\left|sin\ x\right|\ +\ C \)

15. \(\int{sec\ x\ .dx\ }\ \)
\( =\ log\left|sec\ x\ +\ tan\ x\right|\ +\ C\ \)

16. \( \int{sec\ x\ .dx\ }\ \)
\(=log\left|\tan{\left(\frac{\pi}{2}\ +\ \frac{x}{2}\right)}\right|\ +\ C\ \)

17. \( \int{cosec\ x\ .dx}\ \)
\(=\ log\left|cosec\ x\ -\ cot\ x\right|\ +\ C\ \)
\(=\ log\left|tan\ \frac{x}{2}\right|\ +\ C \)

Steps to Solve Integration Questions by Substitution Method

  • Select a new variable , say y =g(x)
  • Now after selecting the new variable , find out the value of dx , by using the \( \frac{dy}{dx} \) = g'(x) .This implies
    dy = g'(x)dx
  • Now put the value of given integral in the form of another variable.
  • Also change the value of all variable w.r.t the new assumed variable.
  • Integrate the new formed integral.
  • Replace the value of assumed variable with the original one.

Read Also : Trigonometry Formulas

Integration by Partial Fractions

Any rational number of the the form \( \frac{P\left(x\right)}{Q\left(x\right)} \) can by solved by the converting them into some standard partial fraction form . In this method we have to convert the \(\frac{P\left(x\right)}{Q\left(x\right)} \) to a partial fraction form. Some standard Result are shown in the table below.

Sr. NoForm of the Rational NumberPartial Fraction Form
1\( \frac{px\ +\ q}{\left(x-a\right)\left(x-b\right)} \) \( \frac{A}{\left(x-a\right)}\ +\ \frac{B}{\left(x-b\right)}\ \)
2\( \frac{px\ +q}{\left(x-a\right)^2} \)\( \frac{A}{\left(x-a\right)}\ +\ \frac{B}{\left(x-a\right)^2} \)
3\( \frac{px^2+qx+\ r}{\left(x-a\right)\left(x-b\right)\left(x-c\right)} \)\( \frac{A}{\left(x-a\right)}\ +\ \frac{B}{\left(x-b\right)}\ +\ \frac{C}{\left(x-c\right)} \)
4\( \frac{px^2+qx+\ r}{\left(x-a\right)^2\left(x-b\right)} \) \( \frac{A}{\left(x-a\right)}\ +\ \frac{B}{\left(x-a\right)^2}\ +\ \frac{C}{\left(x-b\right)}\ \)
5\( \frac{px^2+qx+\ r}{\left(x-a\right)^3\left(x-b\right)} \)\( \frac{A}{\left(x-a\right)}\ +\ \frac{B}{\left(x-a\right)^2}\ +\ \frac{C}{\left(x-a\right)^3}\ +\ \frac{D}{\left(x-b\right)}\)
6\( \frac{px^2+qx+\ r}{(x-a)(x^2+\ bx\ +\ c\ )} \)\( \frac{A}{\left(x-a\right)}\ +\ \frac{Bx+\ C}{x^2\ +\ bx\ +\ c}\ \)
Partial Fraction Integration Formulas

Integration by Parts

When we have to functions which are in multiplication with each other and their integral value is to be find , then we use integration by parts method. The integration by parts method is as follows

\( \int{u.v}\ dx\ =\ u.\int v.dx\ -\int{{\ \frac{du}{dx}\ .\ \int v.dx}}\ .\ dx \)

The ILATE Concept

To integrate the product of the two functions ( u and v ) , we have decide which function is to be taken as u and which one as v , This Can be easily done by ILATE Concepts.
The ILATE stands for –
I – Inverse Trigonometric functions
L – Logarithmic functions
A – Algebraic functions
T – Trigonometric Functions
E – Exponential Functions .

If we have two functions then we have to chose the function ‘u’ which is earlier in the ILATE concept and the other as ‘v’ .

Some Standard Integral Values

1. \( \int{\frac{1}{x^2\ +\ a^2}\ .dx\ }=\ \frac{1}{a}\ \tan^{-1}{\frac{x}{a}}\ +\ C \)

2. \( \int{\frac{1}{x^2\ -\ a^2}\ .dx}\ =\ \frac{1}{2a}\ \log{\left|\frac{x\ -\ a\ }{x\ +\ a}\right|}\ +\ C \)

3. \( \int{\frac{1}{a^2\ -\ x^2}\ .dx\ }\ =\ \frac{1}{2a}\ \log{\left|\frac{a+x}{a-x}\right|}\ +\ C \)

4. \( \int{\frac{1}{\sqrt{x^2\ +\ a^2}}\ .\ dx\ }\ =\ \log{\left|\frac{x\ +\ \sqrt{x^2\ \ +\ a^2}}{a}\right|}\ +\ C\ \)

5. \( \int{\frac{1}{\sqrt{x^2\ -\ a^2}}\ .\ dx\ }\ =\ \log{\left|\frac{x\ +\ \sqrt{x^2\ \ -\ a^2}}{a}\right|}\ +\ C\ \)

6. \( \int{\frac{1}{\sqrt{a^2\ -\ x^2}}\ .\ dx\ }\ =\ \sin^{-1}{\frac{x}{a}\ }\ +\ C\ \)

7. \( \int{\sqrt{a^2\ -\ x^2}\ }.dx\ =\ \frac{x}{2}\ \sqrt{a^2\ -\ x^2}\ +\ \frac{a^2}{2}\ \sin^{-1}{\frac{x}{a}}\ +\ C \)

8. \( \int\sqrt{x^2\ -\ a^2}\ .dx=\ \frac{x}{2}\ \sqrt{x^2\ -\ a^2}\ -\ \frac{a^2}{2}\ \log{\left|x\ +\ \sqrt{x^2\ -\ a^2\ }\right|}+\ C\ \ \)

9. \( \int\sqrt{x^2\ +\ a^2}\ .dx=\ \frac{x}{2}\ \sqrt{x^2\ +\ a^2}\ +\ \frac{a^2}{2}\ \log{\left|x\ +\ \sqrt{x^2\ +\ a^2\ }\right|}+\ C \)

Integration Formulas
Integration Formulas