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人気のある微分積分 >

integral of xsec^3(x^2)

解

\int x sec^{3} (x^{2}) d x

解

\frac{1}{2} (\frac{sec ^{2} ( x ^{2} ) sin ( x ^{2} )}{2} + \frac{1}{2} ln tan (x^{2}) + sec (x^{2})) + C

解答ステップ

\int x sec^{3} (x^{2}) d x

u置換積分法を適用する

= \int \frac{sec ^{3} ( u )}{2} d u

定数を除く:

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

= \frac{1}{2} \cdot \int sec^{3} (u) d u

積分換算を適用する

= \frac{1}{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

代用を戻す

u = x^{2}

= \frac{1}{2} (\frac{sec ^{2} ( x ^{2} ) sin ( x ^{2} )}{2} + \frac{1}{2} ln tan (x^{2}) + sec (x^{2}))

定数を解答に追加する

= \frac{1}{2} (\frac{sec ^{2} ( x ^{2} ) sin ( x ^{2} )}{2} + \frac{1}{2} ln tan (x^{2}) + sec (x^{2})) + C

グラフ

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