Q. 77 Solve given integrals 聽by using... [FREE SOLUTION]

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Chapter 5: Q. 77 (page 465)

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.
$鈭� \frac{x^{4} - 3}{2 + 3 x^{2}} d x$

Short Answer

Expert verified

$\frac{x^{3}}{9} 鈭� \frac{2}{9} x 鈭� \frac{23}{54} \sqrt{6} \tan^{鈭� 1} 鈦� (\frac{\sqrt{6}}{2} x) + C$

Step by step solution

Step1. Given Information

The integral is as follows.

鈭� \frac{x^{4} 鈭� 3}{2 + 3 x^{2}} d x

The objective is to solve the integral.

Step2. Long division

The polynomial long division method is calculated below.

$\begin{aligned} \frac{\frac{1}{3} x^{2} 鈭� \frac{2}{9}}{3 x^{2} + 2 x^{4} 鈭� 3} \\ \frac{x^{4} + \frac{2}{3} x^{2}}{鈭� \frac{2}{3} x^{2} 鈭� 3} \\ \frac{鈭� \frac{2}{3} x^{2} 鈭� \frac{4}{9}}{鈭� \frac{23}{9}} \end{aligned}$

The expression is in the form of $\frac{x^{4} 鈭� 3}{2 + 3 x^{2}} = \frac{x^{2}}{3} 鈭� \frac{2}{9} + \frac{鈭� \frac{23}{9}}{3 x^{2} + 2} .$

Step3. Solution

The integral is solved below.

$\begin{aligned} 鈭� \frac{x^{4} 鈭� 3}{2 + 3 x^{2}} d x \\ = 鈭� (\frac{x^{2}}{3} 鈭� \frac{2}{9} + \frac{鈭� \frac{23}{9}}{3 x^{2} + 2}) d x \\ = 鈭� \frac{x^{2}}{3} d x 鈭� 鈭� \frac{2}{9} d x 鈭� \frac{23}{9} 鈭� \frac{1}{3 x^{2} + 2} d x \\ = \frac{x^{3}}{9} 鈭� \frac{2}{9} x 鈭� \frac{23}{54} \sqrt{6} \tan^{鈭� 1} 鈦� (\frac{\sqrt{6}}{2} x) + C \end{aligned}$

Therefore, the value is $\frac{x^{3}}{9} 鈭� \frac{2}{9} x 鈭� \frac{23}{54} \sqrt{6} \tan^{鈭� 1} 鈦� (\frac{\sqrt{6}}{2} x) + C$

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91影视

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.
$鈭� \frac{x^{4} - 3}{2 + 3 x^{2}} d x$

Short Answer

Step by step solution

Step1. Given Information

Step2. Long division

Step3. Solution

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Discrete Mathematics

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Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.鈭�x4-32+3x2dx

Short Answer

Step by step solution

Step1. Given Information

Step2. Long division

Step3. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Geometry

Applied Mathematics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.
$鈭� \frac{x^{4} - 3}{2 + 3 x^{2}} d x$