Practice Problems for Module 6

Sections 7.4 and 7.8

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1. Evaluate the following integrals.
1. $$\displaystyle \int \! \frac{3x^2+4x+3}{x^2+1} \, dx$$

$$\displaystyle \int \! \frac{3x^2+4x+3}{x^2+1} \, dx = 3x+2\ln|x^2+1|+C$$

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2. $$\displaystyle \int \! \frac{x^2-4x+2}{(x+2)^2(x+1)} \, dx$$

$$\displaystyle \int \! \frac{x^2-4x+2}{(x+2)^2(x+1)} \, dx = -6\ln|x+2|+\frac{14}{x+2}+7\ln|x+1|+C$$

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3. $$\displaystyle \int \! \frac{2x^3-4x^2+18x-11}{\left(x^2+9\right)\left(x^2+4\right)} \, dx$$

$$\displaystyle \int \! \frac{2x^3-4x^2+18x-11}{\left(x^2+9\right)\left(x^2+4\right)} \, dx = -\frac{5}{3}\arctan(3x)+\ln\left|x^2+4\right|+\frac{1}{2}\arctan(2x)+C$$

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2. Determine if the following integrals converge or diverge. If it converges, compute the value.
1. $$\displaystyle \int_{-1}^2 \! \frac{9}{(x+1)^2} \, dx$$

Diverges

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2. $$\displaystyle \int_0^\infty \! \frac{5x^2}{\sqrt[3]{x^3+1}} \, dx$$

Diverges

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3. $$\displaystyle \int_1^5 \! (x-1)\ln(x-1) \, dx$$

Converges
$$\displaystyle \int_1^5 \! (x-1)\ln(x-1) \, dx = 8\ln(4)-4$$

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4. $$\displaystyle \int_3^\infty \! \frac{4}{3x^2+4x} \, dx$$

Converges
$$\displaystyle \int_3^\infty \! \frac{4}{3x^2+4x} \, dx=\ln\left(\frac{13}{9}\right)$$

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3. Use the Comparison Test to determine if the following integrals converge or diverge.
1. $$\displaystyle \int_1^\infty \! \frac{3\sin^2(x)+2}{\sqrt{x}} \, dx$$

Diverges

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2. $$\displaystyle \int_2^\infty \! \frac{\cos(x)+5}{\sqrt{x^3+7}} \, dx$$

Converges

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3. $$\displaystyle \int_3^\infty \! \frac{10}{\sqrt{x}+5x\sqrt{x}} \, dx$$

Converges

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4. $$\displaystyle \int_1^\infty \! \frac{2+e^{-x}}{x} \, dx$$

Diverges

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