Evaluate the Integral integral of (2x)e^(4x) with respect to x

Problem

(∫_^)((2*x)*e(4*x)*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 components for the formula by letting u=2*x and d(v)=e(4*x)*d(x)

3. Differentiate u to find d(u)=2*d(x)

4. Integrate d(v) to find v=(∫_^)(e(4*x)*d(x))=1/4*e(4*x)

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

(∫_^)((2*x)*e(4*x)*d(x))=(2*x)*(1/4*e(4*x))−(∫_^)((1/4*e(4*x))*(2)*d(x))

6. Simplify the terms before performing the final integration.

(∫_^)((2*x)*e(4*x)*d(x))=1/2*x*e(4*x)−(∫_^)(1/2*e(4*x)*d(x))

7. Evaluate the remaining integral.

(∫_^)(1/2*e(4*x)*d(x))=1/2⋅1/4*e(4*x)=1/8*e(4*x)

8. Combine the results and add the constant of integration C

1/2*x*e(4*x)−1/8*e(4*x)+C

Final Answer

(∫_^)((2*x)*e(4*x)*d(x))=1/2*x*e(4*x)−1/8*e(4*x)+C

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