Answer. This is a definite integral. One way to handle it is to
find an antiderivative of and then
evaluate the result at 0 and .
Use integration by parts technique. Set

This gives

The technique of integrating rational functions gives

After easy calculations, we get

Now that we have the antiderivative we use it to get

Problem 2. Evaluate

Answer. We will use Substitution Techniques. Set

We have . The new integral is

The technique of integrating rational functions gives

Since

we have

Do not forget to go back to the variable x, we have

Problem 3. Evaluate

Answer. Note that this integral is direct from the formula

Indeed, if we let , then we have

in other words, we have

On the other hand, we may want to use the technique of integrating
rational functions of and . Let us show the main points
of this technique on this example for the sake of being somehow more
complete!!!

Set . Then we have

This gives

Easy algebraic computations yield:

Since

then we have

Back to the variable x, we have

Using the half-angle formula for the cosine we obtain

and thus finally

Problem 4. Evaluate

Answer. If we complete the square we get

which suggests the substitution . Hence . Therefore we have