Question in picture find the integral by completing the square.18. /x²+3xxp²

Name: Question in picture find the integral by completing the square.18. /x²
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Description: Question in picture find the integral by completing the square.18. /x²+3xxp²

Let the integral be, I=∫1x2+3x-5dx Apply the Completing Square Method,…

Answered: 1 -dx x2+3х — 5

Mathematics For Machine Technology

8th Edition

ISBN:9781337798310

Author:Peterson, John.

Publisher:Peterson, John.

Chapter60: Areas Of Triangles

Section: Chapter Questions

Problem 1A: Find the area of this figure. Round your answer to 2 decimal places.

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Question in picture

find the integral by completing the square.
1
8. /
x²+3x
xp²

Expert Solution

Step 1

Let the integral be,

$I = \int \frac{1}{x^{2} + 3 x - 5} d x$

Apply the Completing Square Method,

$\begin{array}{rcl} I & = & \int \frac{1}{{(x)}^{2} + 2 (x) (\frac{3}{2}) + {(\frac{3}{2})}^{2} - {(\frac{3}{2})}^{2} - 5} d x \\ = & \int \frac{1}{{(x + \frac{3}{2})}^{2} - \frac{9}{4} - 5} d x \\ = & \int \frac{1}{{(x + \frac{3}{2})}^{2} - \frac{29}{4}} d x \end{array}$

Substitution:

$x + \frac{3}{2} = t \Rightarrow d x = d t$

Step 2

Then the integral becomes,

$\begin{array}{rcl} I & = & \int \frac{1}{t^{2} - {(\frac{\sqrt{29}}{2})}^{2}} d t \end{array}$

Apply the formula,

$\int \frac{1}{x^{2} - a^{2}} d x = \frac{1}{2 a} \log |\frac{x - a}{x + a}| + C$

Substitute the values,

$\int \frac{1}{t^{2} - {(\frac{\sqrt{29}}{2})}^{2}} d x = \frac{1}{\sqrt{29}} \log |\frac{2 t - \sqrt{29}}{2 t + \sqrt{29}}| + C$

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