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回編集
→積分領域が変数に依存し、変数変換する必要がない場合
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(1) <math>\int_{0}^{1} \int_{0}^{-x^{2} + 1} xe^{-y} \, dxdy</math> | (1) <math>\int_{0}^{1} \int_{0}^{-x^{2} + 1} xe^{-y} \, dxdy</math> | ||
(2) <math>\int_{0}^{1} int_{x^{2}}^{x} (x + y)^{2} \, dxdy</math> | (2) <math>\int_{0}^{1} \int_{x^{2}}^{x} (x + y)^{2} \, dxdy</math> | ||
<br> | <br> | ||
(1)<br> | (1)<br> | ||
| 141行目: | 141行目: | ||
\begin{align} | \begin{align} | ||
\int_{0}^{1} x(-e^{x^{2} - 1} + 1) \, dx &= \int_{0}^{1} x \, dx - \int_{0}^{1} xe^{x^{2} - 1} \, dx \\ | \int_{0}^{1} x(-e^{x^{2} - 1} + 1) \, dx &= \int_{0}^{1} x \, dx - \int_{0}^{1} xe^{x^{2} - 1} \, dx \\ | ||
&= \left [ \frac{1}{2} x^{2} \right ]_{0}^{1} - \int_{0}^{1} xe^{x^{2} - 1} \, dx \\ | &= \left [ \frac{1}{2} x^{2} \right ]_{0}^{1} - \int_{0}^{1} xe^{x^{2} - 1} \, dx \\ | ||
&= \frac{1}{2} - \int_{0}^{1} xe^{x^{2} - 1} \, dx | &= \frac{1}{2} - \int_{0}^{1} xe^{x^{2} - 1} \, dx | ||
| 168行目: | 167行目: | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
\int_{0}^{1} int_{x^{2}}^{x} (x + y)^{2} \, dxdy &= \int_{0}^{1} dx \biggl [ \frac{(x + y)^{3}}{3} \biggr]_{x^{2}}^{x} \\ | \int_{0}^{1} \int_{x^{2}}^{x} (x + y)^{2} \, dxdy &= \int_{0}^{1} dx \biggl [ \frac{(x + y)^{3}}{3} \biggr]_{x^{2}}^{x} \\ | ||
&= \frac{1}{3} \int_{0}^{1} \left \{ 8x^{3}-(x + x^{2})^{3} \right \} \, dx \\ | &= \frac{1}{3} \int_{0}^{1} \left \{ 8x^{3}-(x + x^{2})^{3} \right \} \, dx \\ | ||
&= \frac{1}{3} \int_{0}^{1} (-x^{6} - 3x^{5} - 3x^{4} + 7x^{3}) \, dx | &= \frac{1}{3} \int_{0}^{1} (-x^{6} - 3x^{5} - 3x^{4} + 7x^{3}) \, dx | ||