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Chapter 4: Q. 18 (page 404)

Indefinite integrals: Use integration formulas, algebra, and
educated guess-and-check strategies to find the following
integrals .
$鈭� \frac{1 - x + x^{7}}{x^{2}} d x$

Short Answer

Expert verified

The integral is $(- \frac{1}{x} - I n |x| + \frac{x^{6}}{6} + C)$ .

Step by step solution

01

Step 1. Given information .

Consider the given integral $鈭� \frac{1 - x + x^{7}}{x^{2}} d x$ .

02

Step 2. Using the sum rule .

The sum rule is $鈭� f (x) 卤 g (x) d x = 鈭� f (x) d x + 鈭� g (x) d x$ .

03

Step 3. Find the integral .

$鈬� 鈭� \frac{1 - x + x^{7}}{x^{2}} d x 鈬� 鈭� \frac{1}{x^{2}} - \frac{1}{x} + x^{5} d x 鈬� 鈭� \frac{1}{x^{2}} d x - 鈭� \frac{1}{x} d x + 鈭� x^{5} d x 鈬� \frac{- 1}{x} - I n |x| + \frac{x^{6}}{6} + C$

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Most popular questions from this chapter

Use the graph of f to estimate the values of A(1), A(2), A(3)

Suppose f is a function whose average value on $[- 3, 1]$ is

$- 2$ and whose average rate of change on the same in-

terval is $4$ . Sketch a possible graph for f . Illustrate the

average value and the average rate of change on your

graph of f .

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{{(k + 1)}^{2}}{n^{3} - 1}$

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_{1}^{3} (x + 1)^{2} d x$

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 0}^{25} k = 325$ and ${鈭�}_{k = 3}^{28} {(k - 3)}^{2} = 14, 910$ , find the value of ${鈭�}_{k = 3}^{25} (k^{2} - 5 k + 9)$ .

See all solutions

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