Methods Of Integration

We need to evaluate the integral .

1. Analyze the Integrand:
The integrand is . Notice that the denominator resembles the derivative of . Let’s compute the derivative of :

2. Derivative of :
If , then , and differentiating with respect to :


Since , we have:


Thus, the derivative of is , so:

3. Substitution:
Notice that the denominator of the integrand is , which matches the derivative. Let’s try a substitution. Set:


Then, , and differentiating:


So:


The integrand becomes:

4. Adjust the Limits:
The limits of integration are from to :
- When , .
- When , .
The integral transforms to:

5. Evaluate the Integral:
Now integrate with respect to :


Substitute the limits:

6. Verify:
The antiderivative of is , and substituting back , the antiderivative of the original integrand is , which can be differentiated to confirm:

, which matches the integrand.

Final Answer:
The value of the integral is: