## Definite Integrals

This page definite integrals we are going to see the definition of definite- integral and also example problems using limit.

Definition:

A basic concept of integral calculus is limit. Generally the concept integration is used to find area between curves within certain limit.

Example 1

Evaluate the following Solution:

To solve this problem we have to use substitution method. That is we are going to change the given function from one variable to another variable. So let us consider

t = cos x

differentiating both side with respect to x

dt = - sin x dx

- sin x dx = dt

sin x dx = - dt

We have changed the given function in terms of "t" from the variable x .So we need to change the limits also.

When x = 0                  when x = Π/2

t = cos 0                     t = cos Π/2

t = 1                           t = 0 Example 2

Evaluate the following Solution:

To solve this problem we have to use the trigonometric formula for sin² x.

The formula for sin² x is (1 - cos 2x)/2 In the first step we have applied the trigonometric formula for sin² x. In the second step we have taken 1/2. Now we got integral (1-cos 2x) .If we integrate 1 we will get x and if we integrate cos 2x we will get sin 2x/2. Then we have applied the upper limit first and then lower limit.

Example 3

Evaluate the following Solution:

To solve this problem we have to use substitution method. That is we are going to change the given function from one variable to another variable. So let us consider

t = Sin⁻¹ x

differentiating with respect to x on both sides

dt = 1/√(1-x²) dx

We have changed the given function in terms of "t" from the variable x .So we need to change the limits also.

When x = 0                  when x = 1

t = Sin⁻¹ 0                     t = Sin⁻¹(1)

t = 0                           t = Π/2 Related pages

Quote on Mathematics definite integrals  definite integrals

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