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Chapter 4: Q. 6 (page 361)

Fill in each of the blanks:
(a) $\int d x = x^{6} + C .$
(b) $x^{6}$ is an antiderivative of .
(c) The derivative of $x^{6}$ is .

Short Answer

Expert verified

(a) $\int 6 x^{5} d x = x^{6} + C .$

(b) $x^{6}$ is an antiderivative of $6 x^{5} .$

(c) The derivative of $x^{6}$ is $6 x^{5} .$

Step by step solution

01

Step 1. Given information.

The given incomplete statements are following.

a) $\int d x = x^{6} + C .$

(b) $x^{6}$ is an antiderivative of .

(c) The derivative of $x^{6}$ is .

02

Step 1. Part (a).

derivative of $x^{6} .$

$\int 6 x^{5} d x = 6 (\frac{x^{5 + 1}}{5 + 1}) + C = x^{6} + C = x^{6}$

so $\int 6 x^{5} d x = x^{6} + C .$

03

Step 3. Part (b).

As $\int 6 x^{5} d x = x^{6} + C .$

So $x^{6}$ is an antiderivative of $6 x^{5}$ .

04

Step 4. Part (c)

Derivative of $x^{6} .$

$\frac{d}{d x} x^{6} = 6 x^{6 - 1} = 6 x^{5}$

So derivative of $x^{6}$ is $6 x^{5} .$

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Most popular questions from this chapter

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

(a) A function f for which the signed area between f and the x-axis on [0, 4] is zero, and a different function g for which the absolute area between g and the x-axis on [0, 4] is zero.

(b) A function f whose signed area on [0, 5] is less than its signed area on [0, 3].

(c) A function f whose average value on [−1, 6] is negative while its average rate of change on the same interval is positive.

Use integration formulas to solve each integral in Exercises 21–62. 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

$\int {(x^{3} + 4)}^{2} d x$ .

Show by exhibiting a counterexample that, in general, $\int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x)$ . 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 integration formulas to solve each integral in Exercises 21–62. 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 .

$\int (\frac{2}{3 x} + 1) d x$

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

$\int_{0}^{1} \frac{1}{1 + x^{2}} d x$

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