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Chapter 4: Q. 28 (page 362)

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.
$鈭� (\frac{3}{x^{2}} - 4) d x$

Short Answer

Expert verified

The answer is $\frac{- 3}{x} - 4 x + C$ .

Step by step solution

01

Step 1. Given Information.

The given integral is $鈭� (\frac{3}{x^{2}} - 4) d x .$

02

Step 2. Integration.

On integrating, we get,

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

03

Step 3. Verification.

On differentiating $\frac{- 3}{x} - 4 x + C$ , we get,

$= \frac{3}{x^{2}} - 4$

Hence proved.

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Most popular questions from this chapter

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

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

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{鈭� 3}^{2} 鈥� \frac{2 x}{x^{2} + 5} d x$

Show by exhibiting a counterexample that, in general, $鈭� f (x) g (x) d x 鈮� (鈭� f (x) d x) (鈭� g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

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

Explain why the formula for the integral of $x^{k}$ does not

apply when $k = - 1 .$ What is the integral of $x^{- 1} ?$

See all solutions

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