Integration is the process of finding the antiderivative. The integration of g′(x) with respect to dx is given by
∫ g′(x) dx = g(x) + C, where C is the constant of integration.
The two types of integrals include:
- Definite Integral: An integral with limits namely upper and lower limit without the constant of integration.
- Indefinite integral: An integral without limits and with an arbitrary constant.
This article covers standard integrals, properties of integration, important formulas and examples on integration which helps students to have a deep knowledge on the topic.
Integrals of Rational and Irrational Functions
Integrals of Trigonometric Functions
Integrals of Exponential and Logarithmic Functions
Properties of Integration
Property 5: (i)
⇒ Also Read Definite and Indefinite Integration
Integration of Trigonometric Functions
1. If m –odd put
2. If n odd put
3. If m, n rationales then put
4. If both even then use reduction method
Some useful Substitutions for Irrational Functions
- Form 1 :
- Form 2:
- Form 3:
- Form 4:
Substitute & then, for
Problems on Integration
I = -I
I = 0
as is odd
Find area between and
Linear Differential Equation
Equation of form .
Ar solve by multiplying integrating function
To get final solution of
Ans : Given equation is
which is linear
We use combinations of different terms to get united term
Following terms are useful
Integrating we get
f cont [a, b] & different function.
Q. If where . find
Q. . Find
Q. is equation of straight line. Find it
Hint differentiate twice.
Q. Find points of minima for Ans
This is solved by
Solving by comparing we get
By Parts Integration
Integration of Irrational Algebraic Functions
Put we get
= 0 if
I = -I
I = 0
Q. as is odd
Q. Find area between and in first quadrant.
Area (semicircle) – Area of ΔABC
Optimisation of area for greatest and least values
If area by and between abscissa is find
Differentiating with Labniz equation