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人気のある微分積分 >

integral from-pi/2 to pi/2 of 4cos^6(x)

解

\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 cos^{6} (x) d x

解

\frac{5 π}{4}

+1

十進法表記

3.92699 \dots

解答ステップ

\int_{- \frac{π}{2}}^{\frac{π}{2}} 4 cos^{6} (x) d x

定数を除く:

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

= 4 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{6} (x) d x

積分換算を適用する

= 4 ([\frac{sin ( x ) cos ^{5} ( x )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{4} (x) d x)

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{4} (x) d x = \frac{3 π}{8}

= 4 ([\frac{sin ( x ) cos ^{5} ( x )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{8})

簡素化

4 ([\frac{sin ( x ) cos ^{5} ( x )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{8}) : 4 ([\frac{1}{6} cos^{5} (x) sin (x)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5 π}{16})

= 4 ([\frac{1}{6} cos^{5} (x) sin (x)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5 π}{16})

限度を計算する:

0

= 4 (0 + \frac{5 π}{16})

簡素化

= \frac{5 π}{4}

グラフ

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