Evaluate the Integral

Problem

(∫_^)((arcsin(x))2*d(x))

Solution

1. Apply integration by parts by letting u=(arcsin(x))2 and d(v)=d(x)

2. Calculate the differentials where d(u)=2*arcsin(x)⋅1/√(,1−x2)*d(x) and v=x

3. Substitute into the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

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

4. Apply integration by parts again to the new integral by letting w=arcsin(x) and d(z)=(2*x)/√(,1−x2)*d(x)

5. Calculate the differentials for the second parts where d(w)=1/√(,1−x2)*d(x) and z=−2√(,1−x2) (using u-substitution t=1−x2.

6. Substitute the second parts into the expression.

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

7. Simplify the integrand inside the second integral.

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

8. Evaluate the final integral and combine all terms.

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

9. Combine with the first step and include the constant of integration C

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

10. Distribute the negative sign to reach the final form.

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

Final Answer

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

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