Evaluate the Integral

Problem

(∫_^)(sin(√(,x))*d(x))

Solution

1. Substitute a new variable to simplify the integrand by letting w=√(,x)

2. Differentiate the substitution to find the relationship between d(x) and d(w) where x=w2 implies d(x)=2*w*d(w)

3. Rewrite the integral in terms of w

(∫_^)(2*w*sin(w)*d(w))

4. Apply integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) where u=2*w and d(v)=sin(w)*d(w)

5. Calculate the components for integration by parts: d(u)=2*d(w) and v=−cos(w)

6. Substitute these into the integration by parts formula:

−2*w*cos(w)−(∫_^)(−2*cos(w)*d(w))

7. Evaluate the remaining integral:

−2*w*cos(w)+2*sin(w)+C

8. Back-substitute w=√(,x) to return to the original variable.

Final Answer

(∫_^)(sin(√(,x))*d(x))=−2√(,x)*cos(√(,x))+2*sin(√(,x))+C

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