Question 1

Evaluate the integral $$\int \frac{dx}{\sqrt{16 - 25x^2$}}.$$

Accepted Answer

$$\frac{1}{5} \sin$$^{-1}$$(5x/4) + C$$

Question 2

Which of the following integrals results in an inverse tangent function?

Accepted Answer

$$\int \frac{dx}{a^2$ + x^2$}$$

Question 3

A medical imaging device uses the function $$f(x) = \cosh x to $$model the intensity of a signal. What is the derivative of this function with respect to $$x$$?

Accepted Answer

$$\sinh x$$

Question 4

Evaluate $$\int_{0}^{1} 6x^2$ dx$$ using the properties of definite integrals.

Accepted Answer

$$2$$

Question 5

Which substitution is most appropriate for evaluating $$\int \frac{dx}{\sqrt{4 - 9x^2$}}$$?

Accepted Answer

$$u = 3x$$

Question 6

A chemical reaction rate is modeled by $$f(x) = \$$sinh^2$$ x. $$What is $$\int \$$sinh^2$$ x dx$$?

Accepted Answer

$$\frac{1}{4} \sinh 2x - \frac{x}{2} + C$$

Question 7

Which of the following is the correct general power rule for integration?

Accepted Answer

$$\int [f(x)]^n f'(x) dx = \frac{[f(x)]^{n+1}}{n+1} + C$$

Question 8

Evaluate $$\int_{1}^{3} (3x^2$ - 5x + 2) dx.$$

Accepted Answer

$$10$$

Question 9

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

Accepted Answer

$$\frac{1}{a} \sec$$^{-1}$$ \left( \frac{|u|}{a} \right) + C$$

Question 10

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = e^t \sinh t. $$Find $$\int e^t \sinh t dt.$$

Accepted Answer

$$\frac{1}{4} e$$^{2t}$$ - \frac{1}{2} t + C$$

Question 11

Which of the following is a property of definite integrals?

Accepted Answer

$$\int_a^b [f(x) + g(x)] dx = \int_a^b f(x) dx + \int_a^b g(x) dx$$

Question 12

Evaluate $$\int \sin$$^{1/3}$$ x \cos x dx.$$

Accepted Answer

$$\frac{3}{4} \sin$$^{4/3}$$ x + C$$

Question 13

Which of the following is the correct identity for hyperbolic functions?

Accepted Answer

$$\$$cosh^2$$ x - \$$sinh^2$$ x = 1$$

Question 14

A physicist is analyzing the area under the curve $$f(x) = 4$$ from $$x = -3 to$$ $$x = 5. $$What is $$\int_{-3}^{5} 4 dx$$?

Accepted Answer

$$32$$

Question 15

Which of the following is true about the constant of integration when integrating a sum of functions?

Accepted Answer

The sum of two unknown constants is still an unknown constant, so we write it as a single $$C.$$

Question 16

Evaluate $$\int \cosh x dx.$$

Accepted Answer

$$\sinh x + C$$

Question 17

Which of the following is the correct result for $$\int 	ext{sech}^2 x dx$$?

Accepted Answer

$$	anh x + C$$

Question 18

A definite integral $$\int_a^b f(x) dx is $$bounded by which of the following inequalities if $$m \leq f(x) \leq M$$ for all $$x in$$ $$[a, b]$$?

Accepted Answer

$$m(b-a) \leq \int_a^b f(x) dx \leq M(b-a)$$

Other Integration Concepts and Formulas in Calculus

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