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人気のある微分積分 >

integral of arccos(ax)

解

\int arccos (a x) d x

解

x arccos (a x) - \frac{1 - a ^{2} x ^{2}}{a} + C

解答ステップ

\int arccos (a x) d x

u置換積分法を適用する

= \int \frac{arccos ( u )}{a} d u

定数を除く:

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

= \frac{1}{a} \cdot \int arccos (u) d u

部分積分を適用する

= \frac{1}{a} (u arccos (u) - \int - \frac{u}{1 - u ^{2}} d u)

\int - \frac{u}{1 - u ^{2}} d u = 1 - u^{2}

= \frac{1}{a} (u arccos (u) - 1 - u^{2})

代用を戻す

u = a x

= \frac{1}{a} (a x arccos (a x) - 1 - (a x)^{2})

簡素化

\frac{1}{a} (a x arccos (a x) - 1 - (a x)^{2}) : x arccos (a x) - \frac{1 - a ^{2} x ^{2}}{a}

= x arccos (a x) - \frac{1 - a ^{2} x ^{2}}{a}

定数を解答に追加する

= x arccos (a x) - \frac{1 - a ^{2} x ^{2}}{a} + C

グラフ

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