Evaluate the Integral integral of xe^(x/2) with respect to x

Problem

(∫_^)(x*e(x/2)*d(x))

Solution

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

2. Choose the variables u and d(v) such that u=x and d(v)=e(x/2)*d(x)

3. Differentiate u to find d(u)=d(x)

4. Integrate d(v) to find v=(∫_^)(e(x/2)*d(x))=2*e(x/2)

5. Substitute these values into the integration by parts formula.

(∫_^)(x*e(x/2)*d(x))=x*(2*e(x/2))−(∫_^)(2*e(x/2)*d(x))

6. Evaluate the remaining integral (∫_^)(2*e(x/2)*d(x))=4*e(x/2)

7. Simplify the expression and add the constant of integration C

2*x*e(x/2)−4*e(x/2)+C

Final Answer

(∫_^)(x*e(x/2)*d(x))=2*x*e(x/2)−4*e(x/2)+C

Want more problems? Check here!