[연습문제] 부정적분

1. 다음 적분을 계산하여라.

\((1)\ \displaystyle\int\sqrt{3x+1}dx={2\over9}(3x+1)^{3\over2}\)

\((2)\ \displaystyle\int x(x^2-2)^2dx=\frac{(x^2-2)^3}{6}\)

\((3)\ \displaystyle\int(x^2-2)^2dx=\int(x^4-2x^2+4)dx={x^5\over5}-{2\over3}x^3+4x\)

\((4)\ \displaystyle\int\frac{dx}{5-2x}=-\frac{\ln|5-2x|}{2}\)

\((5)\ \displaystyle\int\frac{\sqrt{x}}{1+x\sqrt{x}}dx={2\over3}\int\frac{dt}{1+t}={2\over3}\ln(1+t)={2\over3}\ln\left(1+x^{3\over2}\right)\)

\((6)\ \displaystyle\int\left(1-{1\over z}\right)^2\frac{dz}{z^2}=-\int(1-t)^2dt=\frac{(1-t)^3}{3}={1\over3}\left(1-{1\over z}\right)^3\)

2. 다음 적분을 계산하여라.

\((1)\ \displaystyle\int\frac{\cos3x}{\sin3x}dx={1\over3}\int\frac{d(\sin3x)}{\sin3x}=\frac{\ln|\sin3x|}{3}\)

\((2)\ \displaystyle\int2\sqrt{7t-13}dt={4\over21}(7t-13)^{3/2}\)

\((3)\ \displaystyle\int(\ln{x}+1)e^{x\ln{x}}dx=\int(\ln{x}+1)x^xdx=\int dt=t=x^x\)

\((4)\ \displaystyle\int\frac{5x-1}{5x^2-2x+1}dx={1\over2}\int\frac{(5x^2-2x+1)'}{5x^2-2x+1}dx=\frac{\ln(5x^2-2x+1)}{2}\)

\((5)\ \displaystyle\int\frac{\tan^4x}{\cos^2x}dx=\int\tan^4xd(\tan{x})=\frac{\tan^5x}{5}\)

\((6)\ \displaystyle\int3^{\sin{x}}\cos{x}dx=\int3^{\sin{x}}d(\sin{x})=\frac{3^{\sin{x}}}{\ln3}\)

\((7)\ \displaystyle\int\frac{e^{\sqrt{x}}}{\sqrt{x}}dx=2\int e^{\sqrt{x}}d(\sqrt{x})=2e^{\sqrt{x}}\)

\((8)\ \begin{split}\ \int\frac{15x^4-2x^2-55x}{3x^2+11}dx&=\int\left(5x^2-19-\frac{55x-209}{3x^2+11}\right)dx\\&={5\over3}x^3-19x-{55\over6}\ln(3x^2+11)+19\sqrt{11\over3}{\rm Tan}^{-1}\left(\sqrt{3\over11}x\right)\end{split}\)

\((9) \begin{split}\end{split}\)

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