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人気のある微分積分 >

integral from 0 to 1 of x^2(1/2+3/2 x^2)

解

\int_{0}^{1} x^{2} (\frac{1}{2} + \frac{3}{2} x^{2}) d x

解

\frac{7}{15}

+1

十進法表記

0.46666 \dots

解答ステップ

\int_{0}^{1} x^{2} (\frac{1}{2} + \frac{3}{2} x^{2}) d x

\frac{1}{2} + \frac{3}{2} x^{2}

の共役で乗じる

\frac{\frac{1}{4} - \frac{9}{4} x ^{4}}{\frac{1}{2} - \frac{3}{2} x ^{2}}

= \int_{0}^{1} x^{2} \frac{\frac{1}{4} - \frac{9}{4} x ^{4}}{\frac{1}{2} - \frac{3}{2} x ^{2}} d x

未定義の (特異) 点を求める:

x = \frac{1}{3}

存在する場合は

b, a < b < c, f (b) =

未定義

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{0}^{\frac{1}{3}} x^{2} \frac{\frac{1}{4} - \frac{9}{4} x ^{4}}{\frac{1}{2} - \frac{3}{2} x ^{2}} d x + \int_{\frac{1}{3}}^{1} x^{2} \frac{\frac{1}{4} - \frac{9}{4} x ^{4}}{\frac{1}{2} - \frac{3}{2} x ^{2}} d x

\int_{0}^{\frac{1}{3}} x^{2} \frac{\frac{1}{4} - \frac{9}{4} x ^{4}}{\frac{1}{2} - \frac{3}{2} x ^{2}} d x = \frac{4}{45 3}

\int_{\frac{1}{3}}^{1} x^{2} \frac{\frac{1}{4} - \frac{9}{4} x ^{4}}{\frac{1}{2} - \frac{3}{2} x ^{2}} d x = \frac{21 3 - 4}{45 3}

= \frac{4}{45 3} + \frac{21 3 - 4}{45 3}

簡素化

\frac{4}{45 3} + \frac{21 3 - 4}{45 3} : \frac{7}{15}

= \frac{7}{15}

グラフ

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