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人気のある微分積分 >

integral from 0 to pi/3 of 8sec^3(x)

解

\int_{0}^{\frac{π}{3}} 8 sec^{3} (x) d x

解

8 (3 + \frac{1}{2} ln (2 + 3))

+1

十進法表記

19.12423 \dots

解答ステップ

\int_{0}^{\frac{π}{3}} 8 sec^{3} (x) d x

定数を除く:

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

= 8 \cdot \int_{0}^{\frac{π}{3}} sec^{3} (x) d x

積分換算を適用する

= 8 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} \cdot \int_{0}^{\frac{π}{3}} sec (x) d x)

\int_{0}^{\frac{π}{3}} sec (x) d x = ln (2 + 3)

= 8 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3))

簡素化

8 ([\frac{sec ^{2} ( x ) sin ( x )}{2}]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3)) : 8 ([\frac{1}{2} sec (x) tan (x)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3))

= 8 ([\frac{1}{2} sec (x) tan (x)]_{0}^{\frac{π}{3}} + \frac{1}{2} ln (2 + 3))

限度を計算する:

3

= 8 (3 + \frac{1}{2} ln (2 + 3))

グラフ

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