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Evaluate
Solution:
Consider the given equation above
This type of integration cannot be integrated by simple integration. We have to use the technique of integration procedures. If we will use the Integration by Parts, the above equation will be more complicated because it contains radical equation. In this type of integration, we have to use the Algebraic Substitution as follows
Let
So that
Substitute the above values to the given equation, we have
but
Therefore, the above equation becomes
Rationalize the denominator so that the radical equation at the denominator will be eliminated as follows
Divide the numerator and denominator by x2, we have
Substitute the values of limits and therefore, the final answer is
Note: If you are getting the area bounded by the curves, then the sign must be expressed in the absolute value. The area is always positive. If you interchange the position of the limits in the proper integral, then the sign of the final answer will be change.