Evaluate the Integral

Problem

(∫_^)(sin(x)*ecos(x)*d(x))

Solution

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

2. Substitute a new variable u for the exponent:

u=cos(x)

3. Differentiate u with respect to x to find the relationship between d(u) and d(x)

d(u)/d(x)=−sin(x)

d(u)=−sin(x)*d(x)

−d(u)=sin(x)*d(x)

4. Rewrite the integral in terms of u by substituting cos(x) with u and sin(x)*d(x) with −d(u)

(∫_^)(eu*(−d(u)))

−(∫_^)(eu*d(u))

5. Integrate the exponential function with respect to u

−eu+C

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

−ecos(x)+C

Final Answer

(∫_^)(sin(x)*ecos(x)*d(x))=−ecos(x)+C

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