Find the Integral arcsin(x)

Problem

(∫_^)(arcsin(x)*d(x))

Solution

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

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

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

4. Substitute these into the integration by parts formula to get x*arcsin(x)−(∫_^)(x/√(,1−x2)*d(x))

5. Apply a usubstitution for the remaining integral by letting w=1−x2 which implies d(w)=−2*x*d(x) or x*d(x)=−1/2*d(w)

6. Evaluate the integral (∫_^)(x/√(,1−x2)*d(x))=−1/2*(∫_^)(w(−1/2)*d(w))=−1/2*(2*w(1/2))=−√(,1−x2)

7. Combine the results and add the constant of integration C

Final Answer

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

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