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

Problem

(∫_^)(ex*sin(x)*d(x))

Solution

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

2. Assign variables for the first application: let u=sin(x) and d(v)=ex*d(x)

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

4. Substitute into the integration by parts formula:

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

5. Apply integration by parts a second time to the new integral (∫_^)(ex*cos(x)*d(x)) let u=cos(x) and d(v)=ex*d(x)

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

7. Substitute these into the second integral:

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

8. Simplify the expression for the second integral:

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

9. Combine the results back into the original equation:

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

10. Distribute the negative sign:

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

11. Add the integral to both sides to solve for (∫_^)(ex*sin(x)*d(x))

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

12. Divide by 2 and add the constant of integration C

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

Final Answer

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

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