$$\displaystyle \text{sech}^2 x$$

Question 1

Which of the following correctly states the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

$$\displaystyle \frac{d}{dx} \int_a^x f(t)\,dt = f(x)$$

Question 2

Evaluate $$\displaystyle \$$int_0$$^1 3x^2$\,dx$$ using the Fundamental Theorem of Calculus, Part 2.

Accepted Answer

$$1$$

Question 3

Suppose $$F(x) is an $$antiderivative of $$f(x). $$Which of the following is always true?

Accepted Answer

$$\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$$

Question 4

Evaluate $$\displaystyle \int 	an x\,dx.$$

Accepted Answer

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

Question 5

Which substitution is most appropriate for evaluating $$\displaystyle \int x \cos(x^2$)\,dx$$?

Accepted Answer

Let $$u = x^2$

Question 6

Given $$\displaystyle \int \frac{2x+1}{x^2$+x}\,dx$$, what is the first step to solve this integral?

Accepted Answer

Decompose into partial fractions

Question 7

Which of the following is the correct definition of $$\sinh x$$?

Accepted Answer

$$\displaystyle \sinh x = \frac{e^x - e$$^{-x}$$}{2}$$

Question 8

Find $$\displaystyle \frac{d}{dx} 	anh x.$$

Accepted Answer

$$\displaystyle 	ext{sech}^2 x$$

Question 9

Which integration technique is most appropriate for $$\displaystyle \int x e^x\,dx$$?

Accepted Answer

Integration by parts

Question 10

Evaluate $$\displaystyle \int \ln x\,dx.$$

Accepted Answer

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

Question 11

Which identity is most useful for integrating $$\displaystyle \int \sin^2$ x\,dx$$?

Accepted Answer

$$\displaystyle \sin^2$ x = \frac{1}{2}(1 - \cos 2x)$$

Question 12

For $$\displaystyle \int \frac{dx}{x^2$ + 4}$$, which substitution is most appropriate?

Accepted Answer

Let $$x = 2 	an 	heta$$

Question 13

Given $$\displaystyle \int \frac{3x^2$ + 2x + 1}{x^2$ + 1}\,dx$$, what should you do first?

Accepted Answer

Use long division

Question 14

Which of the following is the correct partial fraction decomposition for $$\displaystyle \frac{1}{(x-1)^2$}$$?

Accepted Answer

$$\displaystyle \frac{A}{x-1} + \frac{B}{(x-1)^2$}$$

Question 15

Evaluate $$\displaystyle \$$int_1$$^\infty \frac{1}{x^2$}\,dx$$ and determine if it converges or diverges.

Accepted Answer

It converges to $$1$$

Question 16

Which of the following integrals requires you to check for a vertical asymptote at the lower limit?

Accepted Answer

$$\displaystyle \$$int_0$$^1 \frac{1}{\sqrt{x}}\,dx$$

Question 17

A tank is being filled with water at a rate given by $$r(t) = 5t^2$ liters per minute. How much water enters the tank from t=0$ to t=2$ minutes?

Accepted Answer

$$\displaystyle \$$int_0$$^2 5t^2$\,dt = \frac{40}{3}$$ liters

Question 18

Which of the following is the correct antiderivative of $$\cosh x$$?

Accepted Answer

$$\sinh x + C$$

Question 19

Which of the following is a correct application of the chain rule to $$\displaystyle \frac{d}{dx} \int_{a}^{g(x)} f(t)\,dt$$?

Accepted Answer

$$f(g(x)) \cdot g'(x)$$

Calculus II Study Guide: Integration Techniques and Hyperbolic Functions (Ch. 5.1–8.9)

스터디 가이드 - 연습 문제