Evaluate the Integral

Problem

(∫_^)(sin(√(,x))/√(,x)*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, choosing u=√(,x)

2. Calculate the differential d(u) by differentiating u=x(1/2) with respect to x

d(u)/d(x)=1/(2√(,x))

3. Rearrange the differential to solve for d(x) or to match the terms in the integral.

2*d(u)=1/√(,x)*d(x)

4. Substitute the expressions for u and d(u) into the original integral.

(∫_^)(2*sin(u)*d(u))

5. Integrate the trigonometric function with respect to u

−2*cos(u)+C

6. Back-substitute the original variable x using the relation u=√(,x)

−2*cos(√(,x))+C

Final Answer

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

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