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Chapter 5: Q. 38 (page 441)

Calculate each of the integrals in Exercises 17鈥�46. For some integrals you may need to use polynomial long division, partial fractions, factoring or expanding, or the method of completing the square.
$鈭� \frac{16 x^{2} (x^{2} + 1)}{1 - 2 x} d x$

Short Answer

Expert verified

The value of integral is $- 2 x^{4} - \frac{4}{3} x^{3} - 5 x^{2} - 5 x - \frac{5}{2} \ln (|2 x - 1|) + C$ .

Step by step solution

01

Step 1. Given Information.

The given integral is $鈭� \frac{16 x^{2} (x^{2} + 1)}{1 - 2 x} d x$ .

02

Step 2. Calculation.

Rewrite the integral:

$- 16 鈭� \frac{x^{2} (x^{2} + 1)}{2 x - 1} d x$

Perform the long division method:

- 16 鈭� \frac{x^{2} (x^{2} + 1)}{1 - 2 x} d x = - 16 鈭� (\frac{x^{3}}{2} + \frac{x^{2}}{4} + \frac{5 x}{8} + \frac{5}{16} + \frac{5}{16 (2 x - 1)}) d x = - 16 (鈭� \frac{x^{3}}{2} d x + 鈭� \frac{x^{2}}{4} d x + 鈭� \frac{5 x}{8} d x + 鈭� \frac{5}{16} d x + 鈭� \frac{5}{16 (2 x - 1)} d x) = - 16 (\frac{1}{2} (\frac{x^{4}}{4}) + \frac{1}{4} (\frac{x^{3}}{3}) + \frac{5}{8} (\frac{x^{2}}{2}) + \frac{5}{16} x + \frac{5}{16} 鈭� \frac{1}{2 x - 1} d x) = - 2 x^{4} - \frac{4}{3} x^{3} - 5 x^{2} - 5 x - 5 鈭� \frac{1}{2 x - 1} d x

Let $u = 2 x - 1, d u = 2 d x$

$- 2 x^{4} - \frac{4}{3} x^{3} - 5 x^{2} - 5 x - 5 鈭� \frac{1}{2 x - 1} d x = - 2 x^{4} - \frac{4}{3} x^{3} - 5 x^{2} - 5 x - \frac{5}{2} 鈭� \frac{1}{u} d u = - 2 x^{4} - \frac{4}{3} x^{3} - 5 x^{2} - 5 x - \frac{5}{2} \ln (|u|) + C = - 2 x^{4} - \frac{4}{3} x^{3} - 5 x^{2} - 5 x - \frac{5}{2} \ln (|2 x - 1|) + C$

03

Step 4. Conclusion.

The value of integral is $- 2 x^{4} - \frac{4}{3} x^{3} - 5 x^{2} - 5 x - \frac{5}{2} \ln (|2 x - 1|) + C$ .

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Most popular questions from this chapter

Explain why it makes sense to try the trigonometric substitution $x = \sec 鈦� u$ if an integrand involves the expression $\sqrt{x^{2} 鈭� 1}$

Suppose v(x) is a function of x. Explain why the integral

of dv is equal to v (up to a constant).

Suppose $u (x) = x^{2}$ . Calculate and compare the values of the following definite integrals:

role="math" localid="1648786835678" ${鈭�}_{- 1}^{5} u^{2} d u, {鈭�}_{x = - 1}^{x = 5} u^{2} d u and {鈭�}_{u (- 1)}^{u (5)} u^{2} d u$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) An integral with which we could reasonably apply trigonometric substitution with $x = \tan 鈦� u$ .

(b) An integral with which we could reasonably apply trigonometric substitution with $x = 4 \sec 鈦� u$ .

(c) An integral with which we could reasonably apply trigonometric substitution with $x 鈭� 2 = 3 \sin 鈦� u$ .

Consider the integral $鈭� \sin x \cos x d x$ .

(a) Solve this integral by using u-substitution with $u = \sin x$ and $d u = \cos x d x$ .

(b) Solve the integral another way, using u-substitution with $u = \cos x$ and $d u = 鈭� \sin x d x$ .

(c) How must your two answers be related? Use algebra to prove this relationship.

See all solutions

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