# 演習問題

${\int }_{1}^{2}\left({x}^{3}-3{x}^{2}+\frac{1}{\sqrt{x}}\right)dx$

${\int }_{\frac{1}{2}}^{4}\frac{1}{8x+3}dx$

${\int }_{2}^{4}{e}^{\frac{1}{2}x}dx$

${\int }_{\frac{1}{3}}^{2}{8}^{x}dx$

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{2}}sin\left(x+\frac{\pi }{2}\right)dx$

${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{2}}cos\left(x+\frac{\pi }{2}\right)dx$

${\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}{sec}^{2}\left(x+\frac{\pi }{2}\right)dx$

${\int }_{-\frac{\pi }{4}}^{-\frac{\pi }{6}}{\text{cosec}}^{2}\left(x+\frac{\pi }{2}\right)dx$

${\int }_{0}^{\sqrt{3}-1}\frac{1}{{\left(x+1\right)}^{2}+3}dx$

${\int }_{1}^{\sqrt{2}}\frac{1}{\sqrt{4-2{x}^{2}}}dx$

${\int }_{-3}^{0}\frac{1}{\sqrt{{x}^{2}+16}}dx$

${\int }_{0}^{1}{\left(x\sqrt{1-{x}^{2}}\right)}^{3}dx$

${\int }_{0}^{1}\frac{{x}^{2}+1}{x+1}dx$

${\int }_{0}^{\frac{\pi }{4}}\mathrm{sin}x\mathrm{cos}xdx$

${\int }_{1}^{e}x\mathrm{log}xdx$

${\int }_{1}^{3}\frac{4x}{1+{x}^{2}}dx$

${\int }_{\frac{1}{4}}^{1}{16}^{x}dx$

${\int }_{0}^{1}{x}^{2}{e}^{x}dx$

${\int }_{0}^{1}\frac{x}{\sqrt{1-{x}^{2}}}dx$

${\int }_{0}^{1}\frac{x}{{\left(1-{x}^{2}\right)}^{\frac{1}{3}}}dx$

${\int }_{-3}^{-4}x{\left(x+3\right)}^{2}dx$

${\int }_{0}^{1}\sqrt{4-{x}^{2}}dx$

${\int }_{0}^{1}{x}^{2}{e}^{{x}^{3}}dx$

${\int }_{0}^{\sqrt{2}}{x}^{2}\sqrt{4-{x}^{2}}dx$

${\int }_{\frac{1}{2}}^{1}\sqrt{1-{x}^{2}}dx$

${\int }_{2}^{4}xlogxdx$

${\int }_{0}^{\frac{\pi }{2}}xcos2xdx$

${\int }_{0}^{\frac{\pi }{2}}{sin}^{7}xdx$

${\int }_{0}^{\frac{\pi }{2}}{sin}^{4}x{cos}^{2}xdx$

${\int }_{0}^{3}x{e}^{3x}dx$

${\int }_{2}^{5}\frac{1}{2xlog2x}dx$

${\int }_{0}^{\frac{\pi }{4}}\frac{cos2x}{3+sin2x}dx$

${\int }_{0}^{6}|\frac{x}{2}-1|dx$

${\int }_{1}^{3}{4}^{x}dx$

${\int }_{3}^{7}\frac{1}{{x}^{2}-1}dx$

${\int }_{0}^{1}\frac{{x}^{5}}{\sqrt{1-{x}^{2}}}dx$

${\int }_{1}^{2}2dx$

${\int }_{1}^{2}2xdx$

${\int }_{1}^{4}{x}^{2}dx$

$\underset{1}{\overset{4}{\int }}x{e}^{3x}dx$

$y=x+\sqrt{{x}^{2}+5}$逆関数$y=f\left(x\right)$ とする．

（１）$f\left(x\right)$ を求めよ．

（２）定積分 ${\int }_{\sqrt{5}}^{5}f\left(x\right)dx$ を求めよ．

$y=x{e}^{x}\text{ }\left(x\ge 0\right)$逆関数$y=f\left(x\right)$ とおく.