Module 3 Review Problems

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

\int e^{x} \sin (x) d x

$\displaystyle\int {e}^{x}\sin\left(x\right)dx$ cannot be integrated by parts.

2. $\displaystyle\int \frac{1}{{x}^{4}+1}dx$ cannot be integrated using partial fractions.

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3. In numerical integration, increasing the number of points decreases the error.

4. Integration by parts can always yield the integral.

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For the following exercises, evaluate the integral using the specified method.

\int x^{2} \sin (4 x) d x

$\displaystyle\int {x}^{2}\sin\left(4x\right)dx$ using integration by parts

6. $\displaystyle\int \frac{1}{{x}^{2}\sqrt{{x}^{2}+16}}dx$ using trigonometric substitution

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- \frac{\sqrt{x^{2} + 16}}{16 x} + C

$-\frac{\sqrt{{x}^{2}+16}}{16x}+C$

\int \sqrt{x} ln (x) d x

$\displaystyle\int \sqrt{x}\text{ln}\left(x\right)dx$ using integration by parts

8. $\displaystyle\int \frac{3x}{{x}^{3}+2{x}^{2}-5x - 6}dx$ using partial fractions

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\frac{1}{10} (4 ln (2 - x) + 5 ln (x + 1) - 9 ln (x + 3)) + C

$\frac{1}{10}\left(4\text{ln}\left(2-x\right)+5\text{ln}\left(x+1\right)-9\text{ln}\left(x+3\right)\right)+C$

\int \frac{x^{5}}{{(4 x^{2} + 4)}^{\frac{5}{2}}} d x

$\displaystyle\int \frac{{x}^{5}}{{\left(4{x}^{2}+4\right)}^{\frac{5}{2}}}dx$ using trigonometric substitution

10. $\displaystyle\int \frac{\sqrt{4-{\sin}^{2}\left(x\right)}}{{\sin}^{2}\left(x\right)}\cos\left(x\right)dx$ using a table of integrals or a CAS

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- \frac{\sqrt{4 - \sin^{2} (x)}}{\sin (x)} - \frac{x}{2} + C

$-\frac{\sqrt{4-{\sin}^{2}\left(x\right)}}{\sin\left(x\right)}-\frac{x}{2}+C$

For the following exercises, integrate using whatever method you choose.

11.

\int \sin^{2} (x) \cos^{2} (x) d x

$\displaystyle\int {\sin}^{2}\left(x\right){\cos}^{2}\left(x\right)dx$

12. $\displaystyle\int {x}^{3}\sqrt{{x}^{2}+2}dx$

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\frac{1}{15} {(x^{2} + 2)}^{\frac{3}{2}} (3 x^{2} - 4) + C

$\frac{1}{15}{\left({x}^{2}+2\right)}^{\frac{3}{2}}\left(3{x}^{2}-4\right)+C$

13.

\int \frac{3 x^{2} + 1}{x^{4} - 2 x^{3} - x^{2} + 2 x} d x

$\displaystyle\int \frac{3{x}^{2}+1}{{x}^{4}-2{x}^{3}-{x}^{2}+2x}dx$

14. $\displaystyle\int \frac{1}{{x}^{4}+4}dx$

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\frac{1}{16} ln (\frac{x^{2} + 2 x + 2}{x^{2} - 2 x + 2}) - \frac{1}{8} \tan^{- 1} (1 - x) + \frac{1}{8} \tan^{- 1} (x + 1) + C

$\frac{1}{16}\text{ln}\left(\frac{{x}^{2}+2x+2}{{x}^{2}-2x+2}\right)-\frac{1}{8}{\tan}^{-1}\left(1-x\right)+\frac{1}{8}{\tan}^{-1}\left(x+1\right)+C$

15.

\int \frac{\sqrt{3 + 16 x^{4}}}{x^{4}} d x

$\displaystyle\int \frac{\sqrt{3+16{x}^{4}}}{{x}^{4}}dx$

For the following exercises, approximate the integrals using the midpoint rule, trapezoidal rule, and Simpson’s rule using four subintervals, rounding to three decimals.

16. [T] ${\displaystyle\int }_{1}^{2}\sqrt{{x}^{5}+2}dx$

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M_{4} = 3.312, T_{4} = 3.354, S_{4} = 3.326

${\text{M}}_{4}=3.312,{\text{T}}_{4}=3.354,{S}_{4}=3.326$

17. [T]

\int_{0}^{\sqrt{π}} e^{- \sin (x^{2})} d x

${\displaystyle\int }_{0}^{\sqrt{\pi }}{e}^{\text{-}\sin\left({x}^{2}\right)}dx$

18. [T] ${\displaystyle\int }_{1}^{4}\frac{\text{ln}\frac{1}{x}}{x}dx$

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M_{4} = - 0.982, T_{4} = - 0.917, S_{4} = - 0.952

${\text{M}}_{4}=-0.982,{\text{T}}_{4}=-0.917,{S}_{4}=-0.952$

For the following exercises, evaluate the integrals, if possible.

19.

\int_{1}^{\infty} \frac{1}{x^{n}} d x

${\displaystyle\int }_{1}^{\infty }\frac{1}{{x}^{n}}dx$ , for what values of

n

$n$ does this integral converge or diverge?

20. ${\displaystyle\int }_{1}^{\infty }\frac{{e}^{\text{-}x}}{x}dx$

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For the following exercises, consider the gamma function given by $\Gamma\left(a\right)={\displaystyle\int }_{0}^{\infty }{e}^{\text{-}y}{y}^{a - 1}dy$ .

21. Show that

Γ (a) = (a - 1) Γ (a - 1)

$\Gamma\left(a\right)=\left(a - 1\right)\Gamma\left(a - 1\right)$ .

22. Extend to show that

Γ (a) = (a - 1)!

$\Gamma\left(a\right)=\left(a - 1\right)\text{!}$ , assuming

a

$a$ is a positive integer.

The fastest car in the world, the Bugati Veyron, can reach a top speed of 408 km/h. The graph represents its velocity.

This figure has a graph in the first quadrant. It increases to where x is approximately 03:00 mm:ss and then drops off steep. The maximum height of the graph, here the drop occurs is approximately 420 km/h.

23. [T] Use the graph to estimate the velocity every 20 sec and fit to a graph of the form

v (t) = a {exp}^{b x} \sin (c x) + d

$v\left(t\right)=a{\text{exp}}^{bx}\sin\left(cx\right)+d$ . (Hint: Consider the time units.)

24. [T] Using your function from the previous problem, find exactly how far the Bugati Veyron traveled in the 1 min 40 sec included in the graph.

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Answers may vary. Ex:

9.405

$9.405$ km

Techniques of Integration: Supplemental Content

Module 3 Review Problems

Candela Citations