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人気のある微分積分 >

integral from 0 to pi/2 of sin(x)cos(2x)

解

\int_{0}^{\frac{π}{2}} sin (x) cos (2 x) d x

解

- \frac{1}{3}

+1

十進法表記

- 0.33333 \dots

解答ステップ

\int_{0}^{\frac{π}{2}} sin (x) cos (2 x) d x

次の恒等を使用する:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int_{0}^{\frac{π}{2}} \frac{sin ( x + 2 x ) + sin ( x - 2 x )}{2} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} sin (x + 2 x) + sin (x - 2 x) d x

簡素化

sin (x + 2 x) + sin (x - 2 x) : sin (3 x) - sin (x)

= \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} sin (3 x) - sin (x) d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{\frac{π}{2}} sin (3 x) d x - \int_{0}^{\frac{π}{2}} sin (x) d x)

\int_{0}^{\frac{π}{2}} sin (3 x) d x = \frac{1}{3}

\int_{0}^{\frac{π}{2}} sin (x) d x = 1

= \frac{1}{2} (\frac{1}{3} - 1)

簡素化

\frac{1}{2} (\frac{1}{3} - 1) : - \frac{1}{3}

= - \frac{1}{3}

グラフ

人気の例

integral from 0 to 3 of e^{5x}

\int_{0}^{3} e^{5 x} d x

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\int_{- 1}^{0} \frac{1}{x ^{2} - 25} d x

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\int_{1}^{4} 3 x d x

integral from 0 to pi of sin(x)*cos(x)

\int_{0}^{π} sin (x) \cdot cos (x) d x

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