Question 1

Which of the following is the correct formula for the indefinite integral of $$x^n ($$where $$n 
eq -1$$)?

Accepted Answer

$$\int x^n dx = \frac{x$$^{n+1}$$}{n+1} + C$$

Question 2

Evaluate $$\int \frac{1}{x} dx.$$

Accepted Answer

$$\ln|x| + C$$

Question 3

Which formula represents the integration by parts for indefinite integrals?

Accepted Answer

$$\int u dv = u v - \int v du$$

Question 4

Find $$\int e^x dx.$$

Accepted Answer

$$e^x + C$$

Question 5

Evaluate $$\int \cos x dx.$$

Accepted Answer

$$\sin x + C$$

Question 6

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

Accepted Answer

$$\ln|\sec x| + C$$

Question 7

Evaluate $$\int x e^x dx$$ using integration by parts.

Accepted Answer

$$x e^x - e^x + C$$

Question 8

Which of the following is the correct formula for $$\int \sec^2$ x dx$$?

Accepted Answer

$$	an x + C$$

Question 9

Evaluate $$\int \sin x dx.$$

Accepted Answer

$$-\cos x + C$$

Question 10

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = 5 e$$^{-0.2t}$$. What is the total amount of drug absorbed over the interval t = 0$ to t = 10$?

Accepted Answer

$$\$$int_0$$^{10} 5 e$$^{-0.2t}$$ dt = \left[ -25 e$$^{-0.2t}$$ \right]_0$$^{10}$$ = -25 e$$^{-2}$$ + 25$$

Question 11

Which of the following is the correct formula for the definite integral using integration by parts?

Accepted Answer

$$\int_a^b u(x) v'(x) dx = \left[ u(x) v(x) \right]_a^b - \int_a^b v(x) u'(x) dx$$

Question 12

Evaluate $$\int \frac{dx}{\sqrt{a^2$ - x^2$}}$$ for $$a \u003e 0.$$

Accepted Answer

$$\arcsin \left( \frac{x}{a} \right) + C$$

Question 13

A population grows according to $$P(t) = 1000 e$$^{0.05t}$$. What is the total population increase from t = 0$ to t = 20$?

Accepted Answer

$$\$$int_0$$^{20} 1000 e$$^{0.05t}$$ dt = \left[ 20000 e$$^{0.05t}$$ \right]_0$$^{20}$$ = 20000 e$$^{1}$$ - 20000$$

Question 14

Evaluate $$\int \ln x dx$$ using integration by parts.

Accepted Answer

$$x \ln x - x + C$$

Question 15

Which of the following is the correct formula for $$\int \frac{dx}{a^2$ + x^2$}$$?

Accepted Answer

$$\frac{1}{a} \arctan \left( \frac{x}{a} \right) + C$$

Question 16

Evaluate $$\int x \cos x dx$$ using integration by parts.

Accepted Answer

$$x \sin x + \cos x + C$$

Question 17

A medical researcher models the rate of change of a medication in the body as $$R(t) = 2t e$$^{-t}$$. What is the total amount of medication absorbed from t = 0$ to t = 5$?

Accepted Answer

$$\$$int_0$$^5 2t e$$^{-t}$$ dt = \left[ -2t e$$^{-t}$$ - 2 e$$^{-t}$$ \right]_0^5$

Question 18

Which of the following is the correct formula for $$\int \frac{dx}{\sqrt{x^2$ - a^2$}}$$ for $$x \u003e a \u003e 0$$?

Accepted Answer

$$\ln \left| x + \sqrt{x^2$ - a^2$} \right| + C$$

Basic Integration Formulas and Integration by Parts

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