
# 2.6E: Exercises

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## Improper Integrals

Evaluate the following integrals. If the integral is not convergent, answer “divergent.”

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

$$\displaystyle ∫^4_2\frac{dx}{(x−3)^2}$$

divergent

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

$$\displaystyle ∫^∞_0\frac{1}{4+x^2}dx$$

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

$$\displaystyle ∫^2_0\frac{1}{\sqrt{4−x^2}}dx$$

$$\displaystyle \frac{π}{2}$$

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

$$\displaystyle ∫^∞_1\frac{1}{xlnx}dx$$

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

$$\displaystyle ∫^∞_1xe^{−x}dx$$

$$\displaystyle \frac{2}{e}$$

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

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

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

Without integrating, determine whether the integral $$\displaystyle ∫^∞_1\frac{1}{\sqrt{x^3+1}}dx$$ converges or diverges by comparing the function $$\displaystyle f(x)=\frac{1}{\sqrt{x^3+1}}$$ with $$\displaystyle g(x)=\frac{1}{\sqrt{x^3}}$$.

Converges

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

Without integrating, determine whether the integral $$\displaystyle ∫^∞_1\frac{1}{\sqrt{x+1}}dx$$ converges or diverges.

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Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.

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

$$\displaystyle ∫^∞_0e^{−x}cosxdx$$

Converges to $$\displaystyle 1/2$$.

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

$$\displaystyle ∫^∞_1\frac{lnx}{x}dx$$

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

$$\displaystyle ∫^1_0\frac{lnx}{\sqrt{x}}dx$$

$$\displaystyle −4$$

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

$$\displaystyle ∫^1_0lnxdx$$

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

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

$$\displaystyle π$$

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

$$\displaystyle ∫^5_1\frac{dx}{\sqrt{x−1}}$$

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

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

diverges

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

$$\displaystyle ∫^∞_0e^{−x}dx$$

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

$$\displaystyle ∫^∞_0sinxdx$$

diverges

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

$$\displaystyle ∫^∞_{−∞}\frac{e^x}{1+e^{2x}}dx$$

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

$$\displaystyle ∫^1_0\frac{dx}{\sqrt[3]{x}}$$

$$\displaystyle 1.5$$

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

$$\displaystyle ∫^2_0\frac{dx}{x^3}$$

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

$$\displaystyle ∫^2_{−1}\frac{dx}{x^3}$$

diverges

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

$$\displaystyle ∫^1_0\frac{dx}{\sqrt{1−x^2}}$$

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

$$\displaystyle ∫^3_0\frac{1}{x−1}dx$$

diverges

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

$$\displaystyle ∫^∞_1\frac{5}{x^3}dx$$

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

$$\displaystyle ∫^5_3\frac{5}{(x−4)^2}dx$$

diverges

Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.

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

$$\displaystyle ∫^∞_1\frac{dx}{x^2+4x};$$ compare with $$\displaystyle ∫^∞_1\frac{dx}{x^2}$$.

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

$$\displaystyle ∫^∞_1\frac{dx}{\sqrt{x}+1};$$ compare with $$\displaystyle ∫^∞_1\frac{dx}{2\sqrt{x}}$$.

Both integrals diverge.

Evaluate the integrals. If the integral diverges, answer “diverges.”

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

$$\displaystyle ∫^∞_1\frac{dx}{x^e}$$

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

$$\displaystyle ∫^1_0\frac{dx}{x^π}$$

diverges

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

$$\displaystyle ∫^1_0\frac{dx}{\sqrt{1−x}}$$

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

$$\displaystyle ∫^1_0\frac{dx}{1−x}$$

diverges

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

$$\displaystyle ∫^0_{−∞}\frac{dx}{x^2+1}$$

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

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

$$\displaystyle π$$

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

$$\displaystyle ∫^1_0\frac{lnx}{x}dx$$

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

$$\displaystyle ∫^e_0ln(x)dx$$

$$\displaystyle 0.0$$

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

$$\displaystyle ∫^∞_0xe^{−x}dx$$

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

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

$$\displaystyle 0.0$$

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

$$\displaystyle ∫^∞_0e^{−x}dx$$

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Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.

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

$$\displaystyle ∫^9_0\frac{dx}{\sqrt{9−x}}$$

$$\displaystyle 6.0$$

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

$$\displaystyle ∫^1_{−27}\frac{dx}{x^{2/3}}$$

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

$$\displaystyle ∫^3_0\frac{dx}{\sqrt{9−x^2}}$$

$$\displaystyle \frac{π}{2}$$

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

$$\displaystyle ∫^{24}_6\frac{dt}{t\sqrt{t^2−36}}$$

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

$$\displaystyle ∫^4_0xln(4x)dx$$

$$\displaystyle 8ln(16)−4$$

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

$$\displaystyle ∫^3_0\frac{x}{\sqrt{9−x^2}}dx$$

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

Evaluate $$\displaystyle ∫^t_{.5}\frac{dx}{\sqrt{1−x^2}}.$$ (Be careful!) (Express your answer using three decimal places.)

$$\displaystyle 1.047$$

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

Evaluate $$\displaystyle ∫^4_1\frac{dx}{\sqrt{x^2−1}}.$$ (Express the answer in exact form.)

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

Evaluate $$\displaystyle ∫^∞_2\frac{dx}{(x^2−1)^{3/2}}.$$

$$\displaystyle −1+\frac{2}{\sqrt{3}}$$

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

Find the area of the region in the first quadrant between the curve $$\displaystyle y=e^{−6x}$$ and the x-axis.

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

Find the area of the region bounded by the curve $$\displaystyle y=\frac{7}{x^2},$$ the x-axis, and on the left by $$\displaystyle x=1.$$

$$\displaystyle 7.0$$

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

Find the area under the curve $$\displaystyle y=\frac{1}{(x+1)^{3/2}},$$ bounded on the left by $$\displaystyle x=3.$$

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

Find the area under $$\displaystyle y=\frac{5}{1+x^2}$$ in the first quadrant.

$$\displaystyle \frac{5π}{2}$$

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

Find the volume of the solid generated by revolving about the x-axis the region under the curve $$\displaystyle y=\frac{3}{x}$$ from $$\displaystyle x=1$$ to $$\displaystyle x=∞.$$

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

Find the volume of the solid generated by revolving about the y-axis the region under the curve $$\displaystyle y=6e^{−2x}$$ in the first quadrant.

$$\displaystyle 3π$$

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

Find the volume of the solid generated by revolving about the x-axis the area under the curve $$\displaystyle y=3e^{−x}$$ in the first quadrant.

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The Laplace transform of a continuous function over the interval $$\displaystyle [0,∞)$$ is defined by $$\displaystyle F(s)=∫^∞_0e^{−sx}f(x)dx$$ (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F is the set of all real numbers s such that the improper integral converges. Find the Laplace transform F of each of the following functions and give the domain of F.

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

$$\displaystyle f(x)=1$$

$$\displaystyle \frac{1}{s},s>0$$

### Exercise $$\PageIndex{56}$$

$$\displaystyle f(x)=x$$

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### Exercise $$\PageIndex{57}$$

$$\displaystyle f(x)=cos(2x)$$

$$\displaystyle \frac{s}{s^2+4},s>0$$

### Exercise $$\PageIndex{58}$$

$$\displaystyle f(x)=e^{ax}$$

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### Exercise $$\PageIndex{59}$$

Use the formula for arc length to show that the circumference of the circle $$\displaystyle x^2+y^2=1$$ is 2π.

A function is a probability density function if it satisfies the following definition: $$\displaystyle ∫^∞_{−∞}f(t)dt=1$$. The probability that a random variable x lies between a and b is given by $$\displaystyle P(a≤x≤b)=∫^b_af(t)dt.$$

### Exercise $$\PageIndex{60}$$

Show that $$\displaystyle f(x)=\begin{cases}0&ifx<0\\7e^{−7x}&ifx≥0\end{cases}$$ is a probability density function.