Chapter 21 Multiple choice questions

* not completed

. Using integration by parts, find: $y={\displaystyle \int x^3{\text{e}}^{5x}\text{d}x}$.

$y=\frac{{\text{e}}^{5x}}{5}\left(x^3-3x^2\right)+c$ correct incorrect

$y=\frac{{\text{2e}}^{5x}}{125}\left(5x-1\right)+c$ correct incorrect

$y=\frac{{\text{3e}}^{5x}}{25}\left(x^2-2x\right)+c$ correct incorrect

$y=\frac{{\text{e}}^{5x}}{625}\left(125x^3-75x^2+30x-6\right)+c$ correct incorrect

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. Using integration by parts, find: $y={\displaystyle \int x^2\ln 3x\text{d}x}$.

$y=\frac{x^3}{9}\left(\ln 3x-1\right)+c$ correct incorrect

$y=\frac{x^3}{9}\left(3\ln 3x-1\right)+c$ correct incorrect

$y=\frac{x^3}{9}\left(\ln 3x+1\right)+c$ correct incorrect

$y=\frac{x^3}{9}\left(-\ln 3x+1\right)+c$ correct incorrect

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. Using integration by parts, find: $y={\displaystyle \int {\text{e}}^{4x}\sin 3x\text{d}x}$.

$y=\frac{{\text{e}}^{4x}}{25}\left(4\sin 3x-3\cos 3x\right)+c$ correct incorrect

$y=25{\text{e}}^{4x}\left(4\sin 3x-3\cos 3x\right)+c$ correct incorrect

$y=\frac{{\text{e}}^{4x}}{25}\left(3\cos 3x-4\sin 3x\right)+c$ correct incorrect

$y=25{\text{e}}^{4x}\left(3\cos 3x-4\sin 3x\right)+c$ correct incorrect

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. Using integration by substitution, find: $y={\displaystyle \int x^7{\left(2x^8-5\right)}^6\text{d}x}$.

$y=\frac{{\left(2x^8-5\right)}^7}{42}+c$ correct incorrect

$y=\frac{{\left(2x^8-5\right)}^7}{112}+c$ correct incorrect

$y=\frac{x^8{\left(2x^8-5\right)}^7}{42}+c$ correct incorrect

$y=\frac{x^8{\left(2x^8-5\right)}^7}{112}+c$ correct incorrect

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. Using integration by substitution, find: $y={\displaystyle \int \frac{x^2}{4x^3-7}\text{d}x}$.

$y=\ln 4x-\frac{x^3}{14}+c$ correct incorrect

$y=\frac{1}{4x^2-7x}+c$ correct incorrect

$y=\ln \left(4x^3\right)-\frac{3x^3}{7}+c$ correct incorrect

$y=\frac{\ln \left(4x^3-7\right)}{12}+c$ correct incorrect

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. Using integration by substitution, find: $y={\displaystyle \int \frac{\cos \left(\sqrt{x}\right)}{\sqrt{x}}\text{d}x}$.

$y=\sin \left(\sqrt{x}\right)+c$ correct incorrect

$y=2\sin \left(\sqrt{x}\right)+c$ correct incorrect

$y=\frac{1}{\sin \left(\sqrt{x}\right)}+c$ correct incorrect

$y=\frac{2}{\sin \left(\sqrt{x}\right)}+c$ correct incorrect

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. Using integration by backward substitution, find: $y={\displaystyle \int \frac{x}{\sqrt{x+5}}\text{d}x}$.

$y=\sqrt{\left(x+5\right)}+c$ correct incorrect

$y=\ln \left(\sqrt{\left(x+5\right)}\right)+c$ correct incorrect

$y=\frac{2}{3}\sqrt{{\left(x+5\right)}^3}+c$ correct incorrect

$y=\frac{2}{3}\left(\sqrt{{\left(x+5\right)}^3}-15\sqrt{x+5}\right)+c$ correct incorrect

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. Calculate the following integral: $y={\displaystyle \int \frac{1}{x^2+4}}\text{d}x$.

$y=\frac{1}{2}tan\left(2x\right)+c$ correct incorrect

$y=\frac{1}{2}\arctan \left(\frac{x}{2}\right)+c$ correct incorrect

$y=-\frac{1}{2}\cot \left(2x\right)+c$ correct incorrect

$y=\frac{1}{2}\ln \left(x^2+4\right)+c$ correct incorrect

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. Calculate the following integral: $y={\displaystyle {\int }_0^{\infty }{x}^3{\text{e}}^{-4x}}\text{d}x$.

$y=\frac{3}{128}$ correct incorrect

$y=\frac{3}{256}$ correct incorrect

$y=\frac{1}{2}\sqrt{\frac{\pi }{4}}$ correct incorrect

$y=\frac{\sqrt{\pi }}{2}$ correct incorrect

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. Calculate the following integral: $y={\displaystyle {\int }_0^{\frac{\pi }{2}}\sin \left(4x\right)\sin \left(6x\right)}\text{d}x$.

$y=\frac{\pi }{2}$ correct incorrect

$y=\frac{3\pi }{4}$ correct incorrect

$y=\frac{\pi }{4}$ correct incorrect

$y=0$ correct incorrect