Evaluate the Integral

Problem

(∫_^)(cos(x)*esin(x)*d(x))

Solution

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

2. Substitute a new variable u for the exponent.

u=sin(x)

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

d(u)/d(x)=cos(x)

d(u)=cos(x)*d(x)

4. Rewrite the integral in terms of u by substituting u and d(u) into the original expression.

(∫_^)(eu*d(u))

5. Integrate the exponential function with respect to u

eu+C

6. Back-substitute the original expression for u to return the answer in terms of x

esin(x)+C

Final Answer

(∫_^)(cos(x)*esin(x)*d(x))=esin(x)+C

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