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Chapter 4: Problem 38

Evaluate using integration by parts. $$\int_{0}^{\ln 3} x^{2} e^{2 x} d x$$

Short Answer

Expert verified
The integral evaluates to \( \frac{9}{2} (\text{ln} 3)^2 - \frac{9}{4} \text{ln} 3 + \frac{9}{8} \).

Step by step solution

01

- Identify parts for integration by parts

The integration by parts formula is: \[ \text{鈭珆 u dv = uv - \text{鈭珆 v du \]. Let \( u = x^2 \) and \( dv = e^{2x} dx \).
02

- Differentiate u and integrate dv

Calculate \( du \) and \( v \): \[ u = x^2 \rightarrow du = 2x \, dx \]. \[ dv = e^{2x} \, dx \rightarrow v = \frac{1}{2} e^{2x} \].
03

- Apply integration by parts formula

Substitute \( u, v, du, \) and \( dv \) into the formula: \[ \text{鈭珆 x^2 e^{2x} dx = \frac{1}{2} x^2 e^{2x} - \text{鈭珆 \frac{1}{2} e^{2x} \times 2x \, dx \] \[ = \frac{1}{2} x^2 e^{2x} - \text{鈭珆 x e^{2x} dx \].
04

- Apply integration by parts again

Use integration by parts again for \( \text{鈭珆 x e^{2x} dx \). Let \( u = x \) and \( dv = e^{2x} dx \). Then: \[ du = dx \] \[ v = \frac{1}{2} e^{2x} \]. Substitute into the formula: \[ \text{鈭珆 x e^{2x} dx = \frac{1}{2} x e^{2x} - \text{鈭珆 \frac{1}{2} e^{2x} dx \] \[ = \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \].
05

- Substitute back into the original equation

Combine the parts: \[ \text{鈭珆 x^2 e^{2x} dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{2} \big( \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \big) \].
06

- Simplify the expression

Simplify: \[ \text{鈭珆 x^2 e^{2x} dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{4} x e^{2x} + \frac{1}{8} e^{2x} \].
07

- Evaluate definite integral

Evaluate from \(0\) to \( \text{ln} 3 \): \[ \bigg[ \frac{1}{2} x^2 e^{2x} - \frac{1}{4} x e^{2x} + \frac{1}{8} e^{2x} \bigg]_{0}^{\text{ln} 3} \]. At \( x = \text{ln} 3 \): \[ \frac{1}{2} (\text{ln} 3)^2 e^{2 \text{ln} 3} - \frac{1}{4} (\text{ln} 3) e^{2 \text{ln} 3} + \frac{1}{8} e^{2 \text{ln} 3} = \frac{1}{2} (\text{ln} 3)^2 \times 9 - \frac{1}{4} (\text{ln} 3) \times 9 + \frac{1}{8} \times 9 \]. Simplify: \[ \frac{9}{2} (\text{ln} 3)^2 - \frac{9}{4} \text{ln} 3 + \frac{9}{8} \]. At \( x = 0 \), all parts are zero, so the integral evaluates to: \[ \frac{9}{2} (\text{ln} 3)^2 - \frac{9}{4} \text{ln} 3 + \frac{9}{8} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

definite integral
A definite integral represents the signed area under a curve from one point to another. It's written as \(\text{鈭珆_{a}^{b} f(x) \, dx\) where \(a \) and \(b \) are the limits of integration. Unlike an indefinite integral, which deals with antiderivatives and a constant of integration (\

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