Evaluate the Integral integral of e^xcos(x) with respect to x

Problem

(∫_^)(ex*cos(x)*d(x))

Solution

1. Identify the method of integration by parts, where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=cos(x) and d(v)=ex*d(x)

2. Calculate the differentials d(u)=−sin(x)*d(x) and v=ex

3. Apply the formula for the first time:

(∫_^)(ex*cos(x)*d(x))=ex*cos(x)−(∫_^)(ex*(−sin(x))*d(x))

4. Simplify the expression:

(∫_^)(ex*cos(x)*d(x))=ex*cos(x)+(∫_^)(ex*sin(x)*d(x))

5. Apply integration by parts again to the new integral (∫_^)(ex*sin(x)*d(x)) Let u=sin(x) and d(v)=ex*d(x)

6. Calculate the differentials d(u)=cos(x)*d(x) and v=ex

7. Substitute this back into the equation:

(∫_^)(ex*cos(x)*d(x))=ex*cos(x)+(ex*sin(x)−(∫_^)(ex*cos(x)*d(x)))

8. Combine the identical integral terms on the left side of the equation:

2*(∫_^)(ex*cos(x)*d(x))=ex*cos(x)+ex*sin(x)

9. Solve for the integral by dividing by 2 and adding the constant of integration C

Final Answer

(∫_^)(ex*cos(x)*d(x))=(ex*(cos(x)+sin(x)))/2+C

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