Evaluate the Integral

Problem

(∫_^)(x/√(,1−x2)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the numerator x is a multiple of the derivative of the expression inside the square root 1−x2

2. Substitute u=1−x2

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=−2*x

d(u)=−2*x*d(x)

−1/2*d(u)=x*d(x)

4. Rewrite the integral in terms of u

(∫_^)(1/√(,u)*(−1/2)*d(u))

−1/2*(∫_^)(u(−1/2)*d(u))

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

−1/2*((u(1/2))/(1/2))+C

−u(1/2)+C

6. Back-substitute the original expression for u

−√(,1−x2)+C

Final Answer

(∫_^)(x/√(,1−x2)*d(x))=−√(,1−x2)+C

Want more problems? Check here!