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人気のある微分積分 >

integral from 0 to pi/2 of cos^4(t)

解

\int_{0}^{\frac{π}{2}} cos^{4} (t) d t

解

\frac{3 π}{16}

+1

十進法表記

0.58904 \dots

解答ステップ

\int_{0}^{\frac{π}{2}} cos^{4} (t) d t

簡素化

cos^{4} (t) : cos^{3} (t) cos (t)

= \int_{0}^{\frac{π}{2}} cos^{3} (t) cos (t) d t

部分積分を適用する

= [cos^{3} (t) sin (t) - \int - 3 cos^{2} (t) sin^{2} (t) d t]_{0}^{\frac{π}{2}}

\int - 3 cos^{2} (t) sin^{2} (t) d t = - \frac{3}{8} (t - \frac{1}{4} sin (4 t))

= [cos^{3} (t) sin (t) - (- \frac{3}{8} (t - \frac{1}{4} sin (4 t)))]_{0}^{\frac{π}{2}}

簡素化

[cos^{3} (t) sin (t) - (- \frac{3}{8} (t - \frac{1}{4} sin (4 t)))]_{0}^{\frac{π}{2}} : [\frac{1}{32} (32 cos^{3} (t) sin (t) + 3 (4 t - sin (4 t)))]_{0}^{\frac{π}{2}}

= [\frac{1}{32} (32 cos^{3} (t) sin (t) + 3 (4 t - sin (4 t)))]_{0}^{\frac{π}{2}}

限度を計算する:

\frac{3 π}{16}

= \frac{3 π}{16}

グラフ

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