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Chapter 7: Problem 30

Evaluate the following integrals. $$\int x^{2} e^{4 x} d x$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int x^{2} e^{4 x} d x$$. Answer: The integral evaluates to $$\frac{1}{4}x^2 e^{4x} - \frac{1}{8}x e^{4x} + \frac{1}{8} e^{4x} + C$$.

Step by step solution

01

Find du

Differentiate u with respect to x to find du: $$du = \frac{d(x^2)}{dx} = 2x \, dx$$
02

Find v

Integrate dv with respect to x to find v: $$v = \int e^{4x} \, dx = \frac{1}{4} e^{4x} + C$$
03

Apply the integration by parts formula

Use the formula to compute the integral: $$\int x^{2} e^{4 x} d x = uv - \int v \, du$$ $$= x^{2} \big(\frac{1}{4} e^{4x}\big) - \int \big(\frac{1}{4} e^{4x}\big)(2x \, dx)$$
04

Simplify the resulting integral

Simplify the expression: $$= \frac{1}{4}x^2 e^{4x} - \frac{1}{2} \int x e^{4x} \, dx$$ Now we will apply integration by parts again to find the integral $$\int x e^{4x} \, dx$$.
05

Apply integration by parts again

Choose new u and dv as follows: $$u = x$$ $$dv = e^{4x} \, dx$$ Find du: $$du = dx$$ Find v: $$v = \int e^{4x} \, dx = \frac{1}{4} e^{4x} + C$$ Apply the integration by parts formula: $$\int x e^{4x} \, dx = uv - \int v \, du$$ $$= x \big(\frac{1}{4} e^{4x}\big) - \int \big(\frac{1}{4} e^{4x}\big) \, dx$$
06

Simplify and find the remaining integral

Simplify the expression: $$= \frac{1}{4}x e^{4x} - \frac{1}{4} \int e^{4x} \, dx$$ Now find the integral: $$= \frac{1}{4}x e^{4x} - \frac{1}{4} \big(\frac{1}{4} e^{4x} + C\big)$$
07

Substitute back and write the final answer

Substitute the result from Step 5 and 6 back into the expression from Step 4: $$\int x^{2} e^{4 x} d x = \frac{1}{4}x^2 e^{4x} - \frac{1}{2} \big(\frac{1}{4}x e^{4x} - \frac{1}{4} \big(\frac{1}{4} e^{4x} + C\big)\big)$$ Simplify the expression and add a constant of integration: $$\int x^{2} e^{4 x} d x = \frac{1}{4}x^2 e^{4x} - \frac{1}{8}x e^{4x} + \frac{1}{8} e^{4x} + C$$

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