
# 2.3E: Exercises

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## Trigonometric Substitution

Simplify the expressions in exercises 1 - 5 by writing each one using a single trigonometric function.

### Exercise $$\PageIndex{1}$$

$$4−4\sin^2θ$$

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### Exercise $$\PageIndex{2}$$

$$9\sec^2θ−9$$

$$9\sec^2θ−9 \quad = \quad 9\tan^2θ$$

### Exercise $$\PageIndex{3}$$

$$a^2+a^2\tan^2θ$$

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### Exercise $$\PageIndex{4}$$

$$a^2+a^2\sinh^2θ$$

$$a^2+a^2\sinh^2θ \quad = \quad a^2\cosh^2θ$$

### Exercise $$\PageIndex{5}$$

$$16\cosh^2θ−16$$

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Use the technique of completing the square to express each trinomial in exercises 6 - 8 as the square of a binomial.

### Exercise $$\PageIndex{6}$$

$$4x^2−4x+1$$

$$4(x−\frac{1}{2})^2$$

### Exercise $$\PageIndex{7}$$

$$2x^2−8x+3$$

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### Exercise $$\PageIndex{8}$$

$$−x^2−2x+4$$

$$−(x+1)^2+5$$

In exercises 9 - 28, integrate using the method of trigonometric substitution. Express the final answer in terms of the original variable.

### Exercise $$\PageIndex{9}$$

$$\displaystyle ∫\frac{dx}{\sqrt{4−x^2}}$$

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### Exercise $$\PageIndex{10}$$

$$\displaystyle ∫\frac{dx}{\sqrt{x^2−a^2}}$$

$$\displaystyle ∫\frac{dx}{\sqrt{x^2−a^2}} \quad = \quad \ln∣x+\sqrt{−a^2+x^2}∣+C$$

### Exercise $$\PageIndex{11}$$

$$\displaystyle ∫\sqrt{4−x^2}\,dx$$

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### Exercise $$\PageIndex{12}$$

$$\displaystyle ∫\frac{dx}{\sqrt{1+9x^2}}$$

$$\displaystyle ∫\frac{dx}{\sqrt{1+9x^2}} \quad = \quad \tfrac{1}{3}\ln∣\sqrt{9x^2+1}+3x∣+C$$

### Exercise $$\PageIndex{13}$$

$$\displaystyle ∫\frac{x^2\,dx}{\sqrt{1−x^2}}$$

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### Exercise $$\PageIndex{14}$$

$$\displaystyle ∫\frac{dx}{x^2\sqrt{1−x^2}}$$

$$\displaystyle ∫\frac{dx}{x^2\sqrt{1−x^2}} \quad = \quad −\frac{\sqrt{1−x^2}}{x}+C$$

### Exercise $$\PageIndex{15}$$

$$\displaystyle ∫\frac{dx}{(1+x^2)^2}$$

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### Exercise $$\PageIndex{16}$$

$$\displaystyle ∫\sqrt{x^2+9}\,dx$$

$$\displaystyle ∫\sqrt{x^2+9}\,dx \quad = \quad 9\left[\frac{x\sqrt{x^2+9}}{18}+\tfrac{1}{2}\ln\left|\frac{\sqrt{x^2+9}}{3}+\frac{x}{3}\right|\right]+C$$

### Exercise $$\PageIndex{17}$$

$$\displaystyle ∫\frac{\sqrt{x^2−25}}{x}\,dx$$

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### Exercise $$\PageIndex{18}$$

$$\displaystyle ∫\frac{θ^3}{\sqrt{9−θ^2}}\,dθ$$

$$\displaystyle ∫\frac{θ^3dθ}{\sqrt{9−θ^2}}\,dθ \quad = \quad −\tfrac{1}{3}\sqrt{9−θ^2}(18+θ^2)+C$$

### Exercise $$\PageIndex{19}$$

$$\displaystyle ∫\frac{dx}{\sqrt{x^6−x^2}}$$

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### Exercise $$\PageIndex{20}$$

$$\displaystyle ∫\sqrt{x^6−x^8}\,dx$$

$$\displaystyle ∫\sqrt{x^6−x^8}\,dx \quad = \quad \frac{(−1+x^2)(2+3x^2)\sqrt{x^6−x^8}}{15x^3}+C$$

### Exercise $$\PageIndex{21}$$

$$\displaystyle ∫\frac{dx}{(1+x^2)^{3/2}}$$

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### Exercise $$\PageIndex{22}$$

$$\displaystyle ∫\frac{dx}{(x^2−9)^{3/2}}$$

$$\displaystyle ∫\frac{dx}{(x^2−9)^{3/2}} \quad = \quad −\frac{x}{9−\sqrt{9+x^2}}+C$$

### Exercise $$\PageIndex{23}$$

$$\displaystyle ∫\frac{\sqrt{1+x^2}}{x}\,dx$$

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### Exercise $$\PageIndex{24}$$

$$\displaystyle ∫\frac{x^2}{\sqrt{x^2−1}}\,dx$$

$$\displaystyle ∫\frac{x^2}{\sqrt{x^2−1}}\,dx \quad = \quad \tfrac{1}{2}(\ln∣x+\sqrt{x^2−1}∣+x\sqrt{x^2−1})+C$$

### Exercise $$\PageIndex{25}$$

$$\displaystyle ∫\frac{x^2}{x^2+4}\,dx$$

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### Exercise $$\PageIndex{26}$$

$$\displaystyle ∫\frac{dx}{x^2\sqrt{x^2+1}}$$

$$\displaystyle ∫\frac{dx}{x^2\sqrt{x^2+1}} \quad = \quad −\frac{\sqrt{1+x^2}}{x}+C$$

### Exercise $$\PageIndex{27}$$

$$\displaystyle ∫\frac{x^2}{\sqrt{1+x^2}}\,dx$$

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### Exercise $$\PageIndex{28}$$

$$\displaystyle ∫^1_{−1}(1−x^2)^{3/2}\,dx$$

$$\displaystyle ∫^1_{−1}(1−x^2)^{3/2}\,dx \quad = \quad \tfrac{1}{8}\left(x(5−2x^2)\sqrt{1−x^2}+3\arcsin x\right)+C$$

In exercises 29 - 34, use the substitutions $$x=\sinh θ, \, \cosh θ,$$ or $$\tanh θ.$$ Express the final answers in terms of the variable $$x$$.

### Exercise $$\PageIndex{29}$$

$$\displaystyle ∫\frac{dx}{\sqrt{x^2−1}}$$

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### Exercise $$\PageIndex{30}$$

$$\displaystyle ∫\frac{dx}{x\sqrt{1−x^2}}$$

$$\displaystyle ∫\frac{dx}{x\sqrt{1−x^2}} \quad = \quad \ln x−\ln∣1+\sqrt{1−x^2}∣+C$$

### Exercise $$\PageIndex{31}$$

$$\displaystyle ∫\sqrt{x^2−1}\,dx$$

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### Exercise $$\PageIndex{32}$$

$$\displaystyle ∫\frac{\sqrt{x^2−1}}{x^2}\,dx$$

$$\displaystyle ∫\frac{\sqrt{x^2−1}}{x^2}\,dx \quad = \quad −\frac{\sqrt{−1+x^2}}{x}+\ln\left|x+\sqrt{−1+x^2}\right|+C$$

### Exercise $$\PageIndex{33}$$

$$\displaystyle ∫\frac{dx}{1−x^2}$$

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### Exercise $$\PageIndex{34}$$

$$\displaystyle ∫\frac{\sqrt{1+x^2}}{x^2}\,dx$$

$$\displaystyle ∫\frac{\sqrt{1+x^2}}{x^2}\,dx \quad = \quad −\frac{\sqrt{1+x^2}}{x}+\text{arcsinh}\, x+C$$

Use the technique of completing the square to evaluate the integrals in exercises 35 - 39.

### Exercise $$\PageIndex{35}$$

$$\displaystyle ∫\frac{1}{x^2−6x}\,dx$$

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### Exercise $$\PageIndex{36}$$

$$\displaystyle ∫\frac{1}{x^2+2x+1}\,dx$$

$$\displaystyle ∫\frac{1}{x^2+2x+1}\,dx \quad = \quad −\frac{1}{1+x}+C$$

### Exercise $$\PageIndex{37}$$

$$\displaystyle ∫\frac{1}{\sqrt{−x^2+2x+8}}\,dx$$

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### Exercise $$\PageIndex{38}$$

$$\displaystyle ∫\frac{1}{\sqrt{−x^2+10x}}\,dx$$

$$\displaystyle ∫\frac{1}{\sqrt{−x^2+10x}}\,dx \quad = \quad \frac{2\sqrt{−10+x}\sqrt{x}\ln\left|\sqrt{−10+x}+\sqrt{x}\right|}{\sqrt{(10−x)x}}+C$$

### Exercise $$\PageIndex{39}$$

$$\displaystyle ∫\frac{1}{\sqrt{x^2+4x−12}}\,dx$$

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### Exercise $$\PageIndex{40}$$

Evaluate the integral without using calculus: $$\displaystyle ∫^3_{−3}\sqrt{9−x^2}\,dx.$$

$$\displaystyle ∫^3_{−3}\sqrt{9−x^2}\,dx \quad = \quad \frac{9π}{2}$$; area of a semicircle with radius 3

### Exercise $$\PageIndex{41}$$

Find the area enclosed by the ellipse $$\dfrac{x^2}{4}+\dfrac{y^2}{9}=1.$$

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### Exercise $$\PageIndex{42}$$

Evaluate the integral $$\displaystyle ∫\frac{dx}{\sqrt{1−x^2}}$$ using two different substitutions. First, let $$x=\cos θ$$ and evaluate using trigonometric substitution. Second, let $$x=\sin θ$$ and use trigonometric substitution. Are the answers the same?

$$\displaystyle ∫\frac{dx}{\sqrt{1−x^2}} \quad = \quad \arcsin(x)+C$$ is the common answer.

### Exercise $$\PageIndex{43}$$

Evaluate the integral $$\displaystyle ∫\frac{dx}{x\sqrt{x^2−1}}$$ using the substitution $$x=\sec θ$$. Next, evaluate the same integral using the substitution $$x=\csc θ.$$ Show that the results are equivalent.

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### Exercise $$\PageIndex{44}$$

Evaluate the integral $$\displaystyle ∫\frac{x}{x^2+1}\,dx$$ using the form $$\displaystyle ∫\frac{1}{u}\,du$$. Next, evaluate the same integral using $$x=\tan θ.$$ Are the results the same?

$$\displaystyle ∫\frac{x}{x^2+1}\,dx \quad = \quad \frac{1}{2}\ln(1+x^2)+C$$ is the result using either method.

### Exercise $$\PageIndex{45}$$

State the method of integration you would use to evaluate the integral $$\displaystyle ∫x\sqrt{x^2+1}\,dx.$$ Why did you choose this method?

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### Exercise $$\PageIndex{46}$$

State the method of integration you would use to evaluate the integral $$\displaystyle ∫x^2\sqrt{x^2−1}\,dx.$$ Why did you choose this method?

Use trigonometric substitution. Let $$x=\sec(θ).$$

### Exercise $$\PageIndex{47}$$

Evaluate $$\displaystyle ∫^1_{−1}\frac{x}{x^2+1}\,dx$$

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### Exercise $$\PageIndex{48}$$

Find the length of the arc of the curve over the specified interval: $$y=\ln x,\quad [1,5].$$ Round the answer to three decimal places.

$$s = 4.367$$ units

### Exercise $$\PageIndex{49}$$

Find the surface area of the solid generated by revolving the region bounded by the graphs of $$y=x^2,\, y=0,\, x=0$$, and $$x=\sqrt{2}$$ about the $$x$$-axis. (Round the answer to three decimal places).

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### Exercise $$\PageIndex{50}$$

The region bounded by the graph of $$f(x)=\dfrac{1}{1+x^2}$$ and the $$x$$-axis between $$x=0$$ and $$x=1$$ is revolved about the $$x$$-axis. Find the volume of the solid that is generated.

$$V = \left(\frac{π^2}{8}+\frac{π}{4}\right) \, \text{units}^3$$

In exercises 51 - 52, solve the initial-value problem for $$y$$ as a function of $$x$$.

### Exercise $$\PageIndex{51}$$

$$(x^2+36)\dfrac{dy}{dx}=1, \quad y(6)=0$$

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### Exercise $$\PageIndex{52}$$

$$(64−x^2)\dfrac{dy}{dx}=1, \quad y(0)=3$$

$$y=\tfrac{1}{16}\ln\left|\dfrac{x+8}{x−8}\right|+3$$

### Exercise $$\PageIndex{53}$$

Find the area bounded by $$y=\dfrac{2}{\sqrt{64−4x^2}},\, x=0,\, y=0$$, and $$x=2$$.

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### Exercise $$\PageIndex{54}$$

An oil storage tank can be described as the volume generated by revolving the area bounded by $$y=\dfrac{16}{\sqrt{64+x^2}},\, x=0,\, y=0,\, x=2$$ about the $$x$$-axis. Find the volume of the tank (in cubic meters).

$$V = 24.6$$ m3

### Exercise $$\PageIndex{55}$$

During each cycle, the velocity $$v$$ (in feet per second) of a robotic welding device is given by $$v=2t−\dfrac{14}{4+t^2}$$, where $$t$$ is time in seconds. Find the expression for the displacement $$s$$ (in feet) as a function of $$t$$ if $$s=0$$ when $$t=0$$.

### Exercise $$\PageIndex{56}$$
Find the length of the curve $$y=\sqrt{16−x^2}$$ between $$x=0$$ and $$x=2$$.
$$s = \frac{2π}{3}$$ units