Evaluate the Integral integral of xarcsin(x) with respect to x

Problem

(∫_^)(x*arcsin(x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign variables for integration by parts by letting u=arcsin(x) and d(v)=x*d(x)

3. Differentiate u to find d(u)=1/√(,1−x2)*d(x) and integrate d(v) to find v=(x2)/2

4. Substitute these into the integration by parts formula:

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

5. Rewrite the remaining integral by adding and subtracting 1 in the numerator to facilitate simplification:

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

6. Split the integral into two parts:

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

7. Evaluate the standard integrals using the formulas (∫_^)(√(,a2−x2)*d(x))=x/2√(,a2−x2)+(a2)/2*arcsin(x/a) and (∫_^)(1/√(,1−x2)*d(x))=arcsin(x)

−1/2*(x/2√(,1−x2)+1/2*arcsin(x))+1/2*arcsin(x)

8. Simplify the expression:

−(x√(,1−x2))/4−1/4*arcsin(x)+1/2*arcsin(x)=−(x√(,1−x2))/4+1/4*arcsin(x)

9. Combine all parts back into the original equation:

(x2)/2*arcsin(x)−(−(x√(,1−x2))/4+1/4*arcsin(x))+C

10. Factor the result to reach the final form:

(2*x2−1)/4*arcsin(x)+(x√(,1−x2))/4+C

Final Answer

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

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