Evaluate the Integral

Problem

(∫_^)(cos(√(,t))*d(t))

Solution

1. Substitute a new variable to simplify the square root by letting u=√(,t)

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

3. Rewrite the integral in terms of u

(∫_^)(2*u*cos(u)*d(u))

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

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

6. Substitute these into the integration by parts formula:

2*u*sin(u)−(∫_^)(2*sin(u)*d(u))

7. Evaluate the remaining integral:

2*u*sin(u)−(−2*cos(u))+C

8. Simplify the expression:

2*u*sin(u)+2*cos(u)+C

9. Back-substitute u=√(,t) to return to the original variable.

Final Answer

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

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