Evaluate the Integral

Problem

(∫_^)(arcsin(x)/√(,1−x2)*d(x))

Solution

1. Identify a suitable substitution by noticing that the derivative of arcsin(x) is 1/√(,1−x2) which is present in the integrand.

2. Substitute u=arcsin(x) which implies that the differential d(u)=1/√(,1−x2)*d(x)

3. Rewrite the integral in terms of u by replacing arcsin(x) with u and 1/√(,1−x2)*d(x) with d(u)

(∫_^)(u*d(u))

4. Integrate with respect to u using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

(u2)/2+C

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

((arcsin(x))2)/2+C

Final Answer

(∫_^)(arcsin(x)/√(,1−x2)*d(x))=((arcsin(x))2)/2+C

Want more problems? Check here!