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人気のある微分積分 >

integral of (tan^3(x))/(sqrt(sec(x)))

解

\int \frac{tan ^{3} ( x )}{sec ( x )} d x

解

\frac{2}{3} sec^{\frac{3}{2}} (x) + \frac{2}{sec ( x )} + C

解答ステップ

\int \frac{tan ^{3} ( x )}{sec ( x )} d x

u置換積分法を適用する

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

拡張

\frac{u ^{2} - 1}{u u} : u - \frac{1}{u u}

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int u d u = \frac{2}{3} u^{\frac{3}{2}}

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

= \frac{2}{3} u^{\frac{3}{2}} - (- \frac{2}{u})

代用を戻す

u = sec (x)

= \frac{2}{3} sec^{\frac{3}{2}} (x) - (- \frac{2}{sec ( x )})

簡素化

= \frac{2}{3} sec^{\frac{3}{2}} (x) + \frac{2}{sec ( x )}

定数を解答に追加する

= \frac{2}{3} sec^{\frac{3}{2}} (x) + \frac{2}{sec ( x )} + C

グラフ

人気の例

integral of-u^2

\int - u^{2} d u

integral from 0 to 7 of 1/(u^2-49)

\int_{0}^{7} \frac{1}{u ^{2} - 49} d u

derivative of (2000/(x^2))

\frac{d}{d x} (\frac{2000}{x ^{2}})

(dx)/(dt)=x^2t^2

\frac{d x}{d t} = x^{2} t^{2}

derivative of (x^3+1/x)

\frac{d}{d x} (\frac{x ^{3} + 1}{x})