Evaluate the Integral

Problem

(∫_0^1)(2*x*ex*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function 2*x and a transcendental function ex use integration by parts.

2. Set up the variables for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=2*x and d(v)=ex*d(x)

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

4. Apply the formula for integration by parts.

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

5. Evaluate the remaining integral.

(∫_^)(2*x*ex*d(x))=2*x*ex−2*ex

6. Apply the limits of integration from 0 to 1

(∫_0^1)(2*x*ex*d(x))=[2*x*ex−2*ex]10

7. Substitute the upper limit x=1 and the lower limit x=0

(2*(1)*e1−2*e1)−(2*(0)*e0−2*e0)

8. Simplify the resulting expression.

(2*e−2*e)−(0−2*(1))

0−(−2)=2

Final Answer

(∫_0^1)(2*x*ex*d(x))=2

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