Question

Use the substitution method to transform the integrand into a rational function of t and then evaluate the function.

Answer 1

Step 1 :

The integral is .

Evaluate the integral by substitution.

Let , then

If , then .

Step 2 :

Substitute corresponding values in .

Step 3 :

Solve the above integral by using partial fractions.

Partial fraction decomposition.

Equate t coefficients and equate constants.

Multiply equation (2) by 2 and then subtract two equations.

Substitute in equation (2).

Substitute and in equation (1).

Step 4 :

Apply formula : .

Substitute .

.

Solution :

.