Q: Which substitution is most appropriate for evaluating $$\int \sqrt{7 - 36x^2$} \, dx$$ using trigonometric substitution?

$$x = \frac{\sqrt{7}}{6} \sin \theta$$

Q: Evaluate $$\int \frac{5x - 5}{(x + 3)(x^2$ + 1)} \, dx$$ using partial fractions.

$$-2 \ln|x + 3| + \ln(x^2$ + 1) - \tan$$^{-1}$$ x + c$$

Question 1

Which of the following is the correct result for the integral $$\int 4t^2$ e$$^{-t}$$ \, dt$$ using integration by parts?

Accepted Answer

$$-4t^2$$ $$e^{-t} - 8t$$ $$e^{-t}$$ - $$8e^{-t} + c$

Question 2

Evaluate $$\int \cos$$^{-1}$$(5x) \, dx$$ using integration by parts.

Accepted Answer

$$x \cos$$^{-1}$$(5x) - \frac{1}{5} \sqrt{1 - 25x^2$} + c$$

Question 3

Which substitution is most appropriate for evaluating $$\int \sqrt{7 - 36x^2$} \, dx$$ using trigonometric substitution?

Accepted Answer

$$x = \frac{\sqrt{7}}{6} \sin 	heta$$

Question 4

Find $$\int \sec$$^{15}$$x 	an^5$ x \, dx.$$

Accepted Answer

$$\frac{1}{19} \sec$$^{19}$$x - \frac{2}{17} \sec$$^{17}$$x + \frac{1}{15} \sec$$^{15}$$x + c$$

Question 5

A pharmaceutical company is modeling the concentration of a drug in the bloodstream as $$C(x) = \cos^2$ x \sin x. $$What is the indefinite integral $$\int \cos^2$ x \sin x \, dx$$?

Accepted Answer

$$-\frac{1}{3} \cos^3$ x + c$$

Question 6

Evaluate $$\int \csc^5$ x \cot^5$ x \, dx.$$

Accepted Answer

$$-\frac{1}{9} \csc^9$ x + \frac{2}{7} \csc^7$ x - \frac{1}{5} \csc^5$ x + c$$

Question 7

A physicist is integrating $$\int x^3$ \sqrt{1 + 36x^2$} \, dx to $$find the moment of inertia of a rod. What is the correct result?

Accepted Answer

$$\frac{1}{6480} (1 + 36x^2$)^{5/2}$$ - \frac{1}{3888} (1 + $$36x^2$$)^{3/2}$$ + c$$

Question 8

Find the integral $$\int \frac{x^3$ + 3x}{x + 1} \, dx by $$dividing and integrating.

Accepted Answer

$$\frac{1}{3} x^3$ - \frac{1}{2} x^2$ + 4x - 4 \ln|x + 1| + c$$

Question 9

Evaluate $$\int \frac{5x - 5}{(x + 3)(x^2$ + 1)} \, dx$$ using partial fractions.

Accepted Answer

$$-2 \ln|x + 3| + \ln(x^2$ + 1) - 	an$$^{-1}$$ x + c$$

Question 10

A chemical engineer needs to integrate $$\int \frac{16x + 60}{x^2$ + 8x + 15} \, dx to $$determine the total reactant over time. What is the correct result?

Accepted Answer

$$10 \ln|x + 5| + 6 \ln|x + 3| + c$$

Question 11

Which integration technique is most appropriate for $$\int 5x^2$ \sin x \, dx$$?

Accepted Answer

Integration by parts, repeated twice

Question 12

Evaluate $$\int 5x^2$ \sin x \, dx.$$

Accepted Answer

$$-5x^2$$ \cos x + 10x \sin x + 10 \cos x + c$$

Question 13

Find $$\int \cos^8$ x \sin^5$ x \, dx.$$

Accepted Answer

$$-\frac{1}{9} \cos^9$ x + \frac{2}{11} \cos$$^{11}$$ x - \frac{1}{13} \cos$$^{13}$$ x + c$$

Question 14

A biologist is using Simpson's Rule with $$n=6 to $$approximate $$\$$int_1$$^2 \frac{\sin x}{x} \, dx. $$What is the correct approximation rounded to six decimal places?

Accepted Answer

0.659330

Question 15

A financial analyst uses the Trapezoidal Rule with $$n=8 to $$estimate $$\$$int_1$$^5 \frac{\ln x}{3 + x} \, dx. $$What is the correct approximation rounded to six decimal places?

Accepted Answer

0.635171

Question 16

Which of the following integrals is best solved using trigonometric substitution?

Accepted Answer

$$\int \sqrt{7 - 36x^2$} \, dx$$

Question 17

A student is asked to integrate $$\int \sin(\ln 4x) \, dx$$ using integration by parts. What is the correct answer?

Accepted Answer

$$\frac{1}{2} [x \sin(\ln 4x) - x \cos(\ln 4x)] + c$$

Step-by-Step Calculus Integration Guidance: MTH 141 Checkpoint 2

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