Q: Which of the following is NOT a basic integration rule listed in the study materials?

$$\int \tan u du = \sin u + C$$

Question 1

Which of the following integrals can be evaluated using the formula $$\int \frac{du}{\sqrt{a^2$-u^2$$}} = \arcsin \frac{u}{a} + C$?

Accepted Answer

$$\int \frac{dx}{\sqrt{4-x^2$}}$$

Question 2

Evaluate $$\int \frac{dx}{x^2$-4x+7} by $$first completing the square in the denominator.

Accepted Answer

$$\frac{1}{\sqrt{3}}\arctan\left(\frac{x-2}{\sqrt{3}}\right) + C$$

Question 3

Which substitution is most appropriate for evaluating $$\int \frac{dx}{2+9x^2$}$$?

Accepted Answer

$$u = 3x$$

Question 4

A medical imaging device uses the function $$f(x) = \frac{1}{\sqrt{1-x^2$}} to $$model a signal. What is the antiderivative of $$f(x)$$?

Accepted Answer

$$\arcsin x + C$$

Question 5

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

Accepted Answer

$$\arcsec|x| + C$$

Question 6

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

Accepted Answer

$$\frac{1}{a}\arctan\frac{u}{a} + C$$

Question 7

A physicist is analyzing the path of a particle and needs to compute $$\int \frac{x\,dx}{\sqrt{x^2$-1}}. $$Which of the following is the correct antiderivative?

Accepted Answer

$$\sqrt{x^2$-1} + C$$

Question 8

Which integration technique is most useful for evaluating $$\int \frac{dx}{x^2$-4x+7}$$?

Accepted Answer

Completing the square

Question 9

Evaluate $$\int \frac{dx}{\sqrt{4-x^2$}}.$$

Accepted Answer

$$\arcsin\left(\frac{x}{2}\right) + C$$

Question 10

A chemical reaction rate is modeled by $$\int \frac{dx}{2+9x^2$}. $$What is the antiderivative?

Accepted Answer

$$\frac{1}{3\sqrt{2}}\arctan\left(\frac{3x}{\sqrt{2}}\right) + C$$

Question 11

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

Accepted Answer

$$\arcsec|x| + C$$

Question 12

A sound engineer is analyzing a signal and needs to compute $$\int \frac{dx}{x^2$-4x+7}. $$What is the first step?

Accepted Answer

Complete the square in the denominator

Question 13

Which of the following is the correct formula for $$\int \frac{du}{u\sqrt{u^2$-a^2$$}}$?

Accepted Answer

$$\frac{1}{a}\arcsec\left|\frac{u}{a}\right| + C$$

Question 14

Evaluate $$\int \frac{dx}{x^2$-1}$$ using basic integration rules.

Accepted Answer

$$\frac{1}{2}\ln\left|\frac{x+1}{x-1}\right| + C$$

Question 15

A jet travels at a speed modeled by $$\int \frac{dx}{x\sqrt{x^2$-1}}. $$Which inverse trigonometric function appears in the solution?

Accepted Answer

$$\arcsec$$

Question 16

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

Accepted Answer

$$\arctan x + C$$

Question 17

A geologist models a canyon cross-section with $$\int \frac{dx}{\sqrt{x^2$-1}}. $$What is the antiderivative?

Accepted Answer

$$\ln|x+\sqrt{x^2$-1}| + C$$

Question 18

Which of the following is NOT a basic integration rule listed in the study materials?

Accepted Answer

$$\int 	an u du = \sin u + C$$

Integration Involving Inverse Trigonometric Functions and Completing the Square

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