Evaluate the Integral

Problem

(∫_^)(ecos(x)*sin(x)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the exponent cos(x) is present in the integrand as −sin(x)

2. Substitute u=cos(x)

3. Differentiate u to find d(u)=−sin(x)*d(x) which implies −d(u)=sin(x)*d(x)

4. Rewrite the integral in terms of u by substituting the expressions found in the previous steps.

(∫_^)(eu*(−d(u)))

5. Factor out the constant −1 from the integral.

−(∫_^)(eu*d(u))

6. Integrate the exponential function with respect to u

−eu+C

7. Back-substitute the original expression for u to return to the variable x

−ecos(x)+C

Final Answer

(∫_^)(ecos(x)*sin(x)*d(x))=−ecos(x)+C

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