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Chapter 5: Q. 60 (page 465)

Solve each of the integrals. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.
$鈭� \frac{x^{3}}{3 x^{2} + 5} d x$ .

Short Answer

Expert verified

The value is $\frac{x^{2}}{6} - \frac{5}{18} \ln (|3 x^{2} + 5|) + C$ .

Step by step solution

01

Step 1. Given Information.

The integral is,

$鈭� \frac{x^{3}}{3 x^{2} + 5} d x$ .

02

Step 2. Solving the integral

$鈭� \frac{x^{3}}{3 x^{2} + 5} d x = 鈭� (\frac{x}{3} - \frac{5 x}{3 (3 x^{2} + 5)}) d x = 鈭� \frac{x}{3} d x - 鈭� \frac{5 x}{3 (3 x^{2} + 5)} d x = \frac{x^{2}}{6} - \frac{5}{3} 鈭� \frac{x}{(3 x^{2} + 5)} d x = \frac{x^{2}}{6} - \frac{5}{3} 鈭� \frac{1}{6 u} d u [u = 3 x^{2} + 5, d u = 6 x d x] = \frac{x^{2}}{6} - \frac{5}{18} 鈭� \frac{1}{u} d u = \frac{x^{2}}{6} - \frac{5}{18} \ln (|u|) + C = \frac{x^{2}}{6} - \frac{5}{18} \ln (|3 x^{2} + 5|) + C$

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Most popular questions from this chapter

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

$(a) 鈭� \frac{4 + x^{2}}{x} d x$ $(b) 鈭� \frac{x}{4 + x^{2}} d x$

role="math" localid="1648759296940" $(c) 鈭� \frac{x^{2}}{4 + x^{2}} d x$ $(d) 鈭� \frac{16 鈭� x^{4}}{4 + x^{2}} d x$

Solve the integral: $鈭� x \sin {(x)}^{2} d x$ .

Explain how to know when to use the trigonometric substitutions $x = a \sin 鈦� u, x = a \tan 鈦� u, a n d x = a \sec 鈦� u$ , Describe the trigonometric identity and the triangle that will be needed in each case. What are the possible values for $x$ and $u$ in each case?

Explain how to use long division to write the improper fraction $\frac{172}{5}$ as the sum of an integer and a proper fraction.

Solve given definite integral.

${鈭�}_{3}^{5} 鈥� \sqrt{x^{2} 鈭� 9} d x$

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