Integration by Completing the Square

Solved Problems

Solution.

Completing the square in the denominator, we get

Make the substitution

Hence

Solution.

The quadratic function in the denominator does not have real roots, so we can't factor it. Therefore, we complete the square:

Express the numerator in terms of $$

Using the table integrals, we get

Solution.

We complete the square in the denominator:

Write the numerator in terms of $$

Hence, we can split the initial integral into two simpler ones:

We calculate both integrals separately.

In the first integral, let $$ Then $$ so that

Using integration formulas from a table of integrals, we can easily evaluate the second integral:

Then

Solution.

Given that

we split the numerator and write the initial integral as the sum of two integrals:

To find the first integral $$ we use the substitution

Then

To evaluate the second integral $$ we complete the square in the denominator:

Now we can express the integral in terms of the inverse sine function:

The final answer is given by

Solution.

We split the numerator and write the initial integral as the sum of two integrals. Notice that

Then

The first integral $$ is solved using the substitution

Hence

To find the second integral $$ we complete the square in the denominator:

Making the change $$ $$ we obtain

So, the initial integral is given by