Evaluate the Integral

Problem

(∫_^)(cos(√(,x))*d(x))

Solution

1. Substitute a new variable to simplify the argument of the cosine function by letting u=√(,x)

2. Differentiate the substitution to find the relationship between d(x) and d(u) where x=u2 implies d(x)=2*u*d(u)

3. Rewrite the integral in terms of u to get (∫_^)(2*u*cos(u)*d(u))

4. Apply integration by parts using the formula (∫_^)(ƒ*g′=ƒ*g−(∫_^)(ƒ′*g)) where ƒ=2*u and g′=cos(u)

5. Calculate the components for integration by parts: ƒ′=2 and g=sin(u)

6. Evaluate the resulting expression 2*u*sin(u)−(∫_^)(2*sin(u)*d(u))

7. Integrate the remaining term to get 2*u*sin(u)+2*cos(u)+C

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

Final Answer

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

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