# 演習問題

$\int 2x\mathrm{log}xdx$

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

$\int 2x{e}^{2x}dx$

$\int xsinxdx$

$\int xcosxdx$

$\int {e}^{x}sinxdx$

$\int \sqrt{{x}^{2}+5}dx$

$\int log2xdx$

$\int log\left(x+1\right)dx$

$\int 2log\left(2x+1\right)dx$

$\int {\left(log2x\right)}^{3}dx$

$\int x\mathrm{log}3xdx$

$\int {\left(\mathrm{log}x\right)}^{2}dx$

$\int x{e}^{x}\mathit{dx}$

$\int {e}^{2x}sinxdx$

$\int sinxcosxdx$

$\int sinxcosxdx$

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

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

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

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

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

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

${\iint }_{D}{e}^{x}\mathrm{sin}ydxdy$     $\left(D:0\leqq x\leqq 1,0\leqq y\leqq \pi x\right)$

${\iint }_{D}\left({x}^{2}+{y}^{2}\right){e}^{-x-y}dxdy$　　$\left(D:-2

フーリエ変換の問題

$f\left(x\right)=\left\{\begin{array}{cc}x& \left(-1

フーリエ変換の問題

$f\left(x\right)=\left\{\begin{array}{cc}\left|x\right|& \left(-1

フーリエ変換の問題

$f\left(x\right)=\left\{\begin{array}{cc}{e}^{x}& \left(x<0\right)\\ {e}^{-x}& \left(x\geqq 0\right)\end{array}\right\$

フーリエ変換の問題

$f\left(x\right)=\left\{\begin{array}{cc}-{e}^{x}& \left(x<0\right)\\ {e}^{-x}& \left(x\geqq 0\right)\end{array}\right\$

( tan 1 y )dy

この式に1が掛けられていると考えて部分積分をする

$\int 1\cdot \left({\mathrm{tan}}^{-1}y\right)dy$$={\int \left(y\right)}^{\prime }\left({\mathrm{tan}}^{-1}y\right)dy$

$=y\left({\mathrm{tan}}^{-1}y\right)-\int y\frac{1}{{y}^{2}+1}dy$

$I=\int {e}^{x}\mathrm{sin}xdx=\int {\left({e}^{x}\right)}^{\prime }\mathrm{sin}xdx$  とおき，部分積分をする．

 $I$ $=\int {\left({e}^{x}\right)}^{\prime }\mathrm{sin}xdx$ $={e}^{x}\mathrm{sin}x-\int {e}^{x}\mathrm{cos}xdx■微分方程式の問題$ $\int \left(4x\mathrm{log}x\right)dx=\int \left\{{\left(2{x}^{2}\right)}^{\prime }\mathrm{log}x\right\}dx$  とおき，部分積分をする． $\int \left\{{\left(2{x}^{2}\right)}^{\prime }\mathrm{log}x\right\}dx$ $=2{x}^{2}\mathrm{log}x-\int 2{x}^{2}\frac{1}{x}dx$