Evaluate the integral. х — 3dxT² +2x + 4)²

Answered: х — 3 dx T² +2x + 4)²

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter6: Fractions

Section6.1: Simplifying Fractions

Problem 31WE

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Question

Evaluate the integral.

Expert Solution

Step 1

Consider the following integral:

$\begin{array}{rcl} \int \frac{x - 3}{{(x^{2} + 2 x + 4)}^{2}} d x & = & \int \frac{\frac{1}{2} (2 x + 2) - 4}{{(x^{2} + 2 x + 4)}^{2}} d x \\ = & \int [\frac{2 x + 2}{2 {(x^{2} + 2 x + 4)}^{2}} - \frac{4}{{(x^{2} + 2 x + 4)}^{2}}] d x \\ = & \int [\frac{x + 1}{{(x^{2} + 2 x + 4)}^{2}} - \frac{4}{{(x^{2} + 2 x + 4)}^{2}}] d x \\ = & \int \frac{x + 1}{{(x^{2} + 2 x + 4)}^{2}} d x - 4 \int \frac{1}{{(x^{2} + 2 x + 4)}^{2}} d x \end{array}$

Step 2

Consider the following equation:

$I_{1} = \begin{array}{rcl} \int \end{array} \begin{array}{rcl} \frac{x + 1}{{(x^{2} + 2 x + 4)}^{2}} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} x \end{array}$

Substitute $u = x^{2} + 2 x + 4 \to \frac{d u}{d x} = 2 x + 2 \to d x = \frac{1}{2 x + 2} d u$ in $I_{1}$ :

$\begin{array}{rcl} I_{1} & = & \int \frac{1}{2 u^{2}} d u \\ = & \frac{1}{2} \times \frac{u^{- 2 + 1}}{- 2 + 1} \\ = & - \frac{1}{2 u} \end{array}$

Substitute $u = x^{2} + 2 x + 4$ :

$I_{1} = - \frac{1}{2 (x^{2} + 2 x + 4)}$

Step 3

Consider the following equation:

$\begin{array}{rcl} I_{2} & = & \int \frac{1}{{(x^{2} + 2 x + 4)}^{2}} d x \\ = & \int \frac{1}{{(x^{2} + 2 x + 1 + 3)}^{2}} d x \\ = & \int \frac{1}{{[{(x + 1)}^{2} + 3]}^{2}} d x \end{array}$

Substitute $x + 1 = u \to d x = d u$ in $I_{2}$ :

$I_{2} = \int \frac{1}{{(u^{2} + 3)}^{2}} d u$

Apply the following reduction formula:

$\int \frac{1}{{(a u^{2} + b)}^{n}} d u = \frac{2 n - 3}{2 b (n - 1)} \begin{array}{rcl} \int \end{array} \frac{1}{{(a u^{2} + b)}^{n - 1}} d u + \frac{u}{2 b (n - 1) {(a u^{2} + b)}^{n - 1}}$

with $a = 1, b = 3, n = 2$

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