Q: Which of the following is the correct result for $$\int (1 + \sin x) / \cos^2$ x \, dx$$?

$$\sec x + \tan x \cdot \sec x + C$$

Q: Which of the following is the correct result for $$\int \frac{1}{a^2$ + x^2$} dx$$?

$$\frac{1}{a} \tan$$^{-1}$$\left(\frac{x}{a}\right) + C$$

Question 1

Which of the following is the correct antiderivative of $$\int x^3$ dx$$?

Accepted Answer

$$\frac{x^4$}{4} + C$$

Question 2

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

Accepted Answer

$$\ln|x| + C$$

Question 3

Which of the following integrals results in $$\sin(x) + C$$?

Accepted Answer

$$\int \cos(x) dx$$

Question 4

Find $$\int \sec(x) 	an(x) dx.$$

Accepted Answer

$$\sec(x) + C$$

Question 5

Evaluate $$\int 	an(x) dx.$$

Accepted Answer

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

Question 6

Let $$a be a $$real number. What is $$\int \frac{1}{\sqrt{a^2$ - x^2$}} dx$$?

Accepted Answer

$$\sin$$^{-1}$$\left(\frac{x}{a}\right) + C$$

Question 7

Evaluate $$\int x e$$^{x^2}$$ dx.$$

Accepted Answer

$$\frac{1}{2} e$$^{x^2}$$ + C$$

Question 8

Given $$u = 2x + 1$$, what is $$\int x \sqrt{2x+1} dx$$?

Accepted Answer

$$\frac{2}{15} (2x+1$$)^{5/2}$$ - \frac{1}{12} (2x+1$$)^{3/2}$$ + C$$

Question 9

Evaluate $$\int \frac{1}{\sqrt{6x - x^2$}} dx.$$

Accepted Answer

$$\sin$$^{-1}$$\left(\frac{x-3}{3}\right) + C$$

Question 10

Evaluate $$\int \frac{8}{x^2$ - 2x + 2} dx.$$

Accepted Answer

$$8 	an$$^{-1}$$(x-1) + C$$

Question 11

Which of the following is the correct result for $$\int (1 + \sin x) / \cos^2$ x \, dx$$?

Accepted Answer

$$\sec x + 	an x \cdot \sec x + C$$

Question 12

Which of the following is the correct formula for the sum $$\sum_{k=1}^{n} k$$?

Accepted Answer

$$\frac{n(n+1)}{2}$$

Question 13

Evaluate $$\sum_{k=0}^{4} k^2$.

Accepted Answer

$$30$$

Question 14

Which substitution is most appropriate for evaluating $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

Let $$u = x^2$

Question 15

A medical researcher is modeling the growth of bacteria with the function $$N(t) = e$$^{kt}$$. Which integral would give the total bacteria count from t=0$ to t=T$?

Accepted Answer

$$\int_{0}^{T} e$$^{kt}$$ dt$$

Question 16

Which of the following is the correct antiderivative for $$\int e$$^{x}$$ dx$$?

Accepted Answer

$$e^{x} + C$

Question 17

A physicist calculates the work done by a variable force $$F(x) = x^2$ from x=0$ to x=2$. Which integral should be used?

Accepted Answer

$$\int_{0}^{2} x^2$ dx$$

Question 18

Which of the following is the correct result for $$\int \frac{1}{a^2$ + x^2$} dx$$?

Accepted Answer

$$\frac{1}{a} 	an$$^{-1}$$\left(\frac{x}{a}ight) + C$$

Question 19

A chemist is analyzing the rate of a reaction given by $$r(t) = \sin(t). $$What is the total change in concentration from $$t=0 to$$ $$t=\pi$$?

Accepted Answer

$$\int_{0}^{\pi} \sin(t) dt$$

Fundamental Integration Techniques and Sigma Notation in Calculus

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