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Evaluate
Solution:
If you examine the given equation, the denominator is complicated because it has a variable and a radical function as well. The differential at the numerator is not match with the differential of the denominator. Therefore, we cannot integrate the given equation by simple integration. If you will integrate the given equation by parts, then the equation will be more complicated and there's no end in integration as well. We have a procedure for that type of integration which is called an algebraic substitution. Let's consider the given equation
Let
Substitute these values to the given equation, we have
But
The equation becomes
Since the first term is indeterminate form, then we have to evaluate the limit of the first term since ∞/∞ is not accepted as a final answer in Mathematics. Let's evaluate the limit of the first term as follows
Apply the L'Hopital's Rule
Therefore,
Rationalize the denominator by multiplying both sides of the fraction by the conjugate of the denominator, we have
