Integration By Partial Fractions

To solve the integral , we need to simplify the expression in the denominator to identify a suitable substitution or method for integration. The quadratic expression can be rewritten by completing the square:

.

Thus, the integral becomes:

.

This resembles the standard integral formula for the inverse tangent function:

.

In our case, implies , and we have replaced by . Applying the formula results in:

, where is the constant of integration.

Therefore, the value of the integral is:

, where is the constant of integration.

Begin with the integral:

Factor the denominator:

Thus, the integral becomes:

Use the substitution , so , which gives .
Substitute and simplify:

Since , we get:

Substitute back for :

The given integral is:

Simplify the integrand:

Substitute this into the integral:

Split the integral:

Let , so and . Substitute into both terms.

For the first term:

For the second term:

Combine the results:

Thus: