Q: Which of the following is the correct power rule for indefinite integrals?

$$\int x^n dx = \frac{x$$^{n+1}$$}{n+1} + C$$, for $$n \neq -1$$

Question 1

Which of the following best describes an antiderivative of a function $$f(x)$$?

Accepted Answer

A function whose derivative is $$f(x).$$

Question 2

Find the antiderivative of $$f(x) = 5x^4$.

Accepted Answer

Question 3

Given $$f(x) = 200$$, what is the general antiderivative $$F(x)$$?

Accepted Answer

$$F(x) = 200x + C$$

Question 4

Find the indefinite integral $$\int 3x^2$ dx.$$

Accepted Answer

$$x^3 + C$

Question 5

Which of the following is the correct power rule for indefinite integrals?

Accepted Answer

$$\int x^n dx = \frac{x$$^{n+1}$$}{n+1} + C$$, for $$n 
eq -1$$

Question 6

Evaluate $$\int 100x$$^{99}$$ dx.$$

Accepted Answer

$$x^{100} + C$

Question 7

Find the indefinite integral $$\int \sin x dx.$$

Accepted Answer

$$-\cos x + C$$

Question 8

Which of the following is the indefinite integral of $$\cos x$$?

Accepted Answer

$$\sin x + C$$

Question 9

A car's acceleration is given by $$a(t) = 5t. If $$its velocity at $$t = 2 is$$ $$v(2) = -20$$, what is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = \frac{5}{2}t^2$ - 30$$

Question 10

Find the indefinite integral $$\int (100x^4$ - 25x - \frac{13}{2}x^2$) dx.$$

Accepted Answer

$$20x^5$$ - \frac{25}{2}$$x^2$$ - \frac{13}{6}$$x^3 + C$

Question 11

Which of the following is the correct indefinite integral for $$\int 	an x dx$$?

Accepted Answer

$$-\ln|\cos x| + C$$

Question 12

Given $$f(x) = 5x^4$ and F(1) = 1$, find the particular antiderivative F(x$$)$$.

Accepted Answer

$$F(x) = x^5$ - 1$$

Question 13

Evaluate $$\int (\sin^4$ x - 2\sin^2$ x) dx.$$

Accepted Answer

$$\frac{1}{5}\sin^5$ x - \frac{2}{3}\sin^3$ x + C$$

Question 14

Which of the following is the indefinite integral of $$\sec^2$ x$$?

Accepted Answer

$$	an x + C$$

Question 15

Find the indefinite integral $$\int 25x^4$ dx.$$

Accepted Answer

$$5x^5 + C$

Question 16

If $$\frac{dy}{dx} = 2x$$ and $$y(0) = 4$$, what is the particular solution for $$y(x)$$?

Accepted Answer

$$y(x) = x^2$ + 4$$

Question 17

Which of the following is the indefinite integral of $$\cot x$$?

Accepted Answer

$$\ln|\sin x| + C$$

Question 18

A medication is administered at a constant rate of $$a(t) = 100 mg/hr. If $$the initial amount in the bloodstream is $$A(0) = 101 mg$$, what is the function $$A(t)$$?

Accepted Answer

$$A(t) = 100t + 101$$

Question 19

Find the indefinite integral $$\int 3x^2$ dx$$ using the power rule.

Accepted Answer

$$x^3 + C$

Question 20

Which of the following is the indefinite integral of $$\frac{1}{x}$$?

Accepted Answer

$$\ln|x| + C$$

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