Methods Of Integration

To solve the integral , we will use the property of definite integrals:

In this problem, . Let us evaluate the integral using the substitution method:

By substituting , we have . The limits of integration change from to to to . Thus, we have:

Changing the limits back, we get:

Adding both integrals and the transformed integral gives:

Simplifying the expression inside the integral:

This simplifies to:

To make further deduction easier, let us substitute , yielding:

Then, the integral:

Now,

The resolved value formation is equivalent to the given answer choices,

Thus, the correct answer is .

To evaluate the integral , we proceed as follows:

We apply a conjugate multiplication trick to simplify the integrand. Consider multiplying the numerator and denominator by the conjugate of the denominator:

This simplifies using the identity , giving:

Split this into two separate integrals:

To solve each part, use the substitution where necessary and integrate:

1. For , substitute , , . The limits change from to , resulting in to :
2. For , substitute , , . The limits change from to , resulting in to :

Now, compute the values:

gives , a\)

Subtract the two results:

Simplifying:

\)

The answer is of the form

Here, , , and

Finally, compute

Therefore, the correct value is 8.