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**

**Integration****Example problems using he above formulas****Substitution method****Decomposition method****Properties of integrals****Integration-by parts****Integration-of Sec³ x****Standard integrals****Integrating quadratic denominator****Integration-using partial fractions**

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