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

Problem

(∫_^)(2*x*e^x*d(x))

Solution

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

2. Choose the variables for substitution by letting u=2*x and d(v)=e^x*d(x)

3. Differentiate u to find d(u)=2*d(x) and integrate d(v) to find v=e^x

4. Substitute these components into the integration by parts formula.

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

5. Simplify the expression by moving the constant outside the integral.

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

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

(∫_^)(2*x*e^x*d(x))=2*x*e^x−2*e^x+C

7. Factor out the common term 2*e^x to simplify the final expression.

(∫_^)(2*x*e^x*d(x))=2*e^x*(x−1)+C

Final Answer

(∫_^)(2*x*e^x*d(x))=2*x*e^x−2*e^x+C

Want more problems? Check here!