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

Problem

(∫_^)(e^x*cos(x)*d(x))

Solution

1. Identify the integration by parts formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=cos(x) and d(v)=e^x*d(x) Then d(u)=−sin(x)*d(x) and v=e^x

2. Apply the formula for the first time.

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

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

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

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

4. Substitute this result back into the original equation.

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

5. Solve for the integral by adding (∫_^)(e^x*cos(x)*d(x)) to both sides.

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

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

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

Final Answer

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

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