Evaluate the Integral

Problem

(∫_^)(x/((1−x2)3)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the derivative of the inner function 1−x2 is proportional to the x in the numerator.

2. Substitute u=1−x2 which implies that d(u)=−2*x*d(x) or x*d(x)=−1/2*d(u)

3. Rewrite the integral in terms of u

(∫_^)(1/(u3)*(−1/2)*d(u))

4. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)+C where n=−3

−1/2*(∫_^)(u(−3)*d(u))=−1/2*((u(−2))/(−2))+C

5. Simplify the resulting expression:

1/(4*u2)+C

6. Back-substitute the original expression for u to return to the variable x

Final Answer

(∫_^)(x/((1−x2)3)*d(x))=1/(4*(1−x2)2)+C

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