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人気のある微分積分 >

integral of (tan(x))^5(sec(x))^3

解

\int (tan (x))^{5} (sec (x))^{3} d x

解

\frac{sec ^{7} ( x )}{7} - \frac{2 sec ^{5} ( x )}{5} + \frac{sec ^{3} ( x )}{3} + C

解答ステップ

\int (tan (x))^{5} (sec (x))^{3} d x

簡素化

= \int tan^{5} (x) sec^{3} (x) d x

三角関数の公式を使用して書き換える

= \int (- 1 + sec^{2} (x))^{2} tan (x) sec^{3} (x) d x

u置換積分法を適用する

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

拡張

u^{2} (u^{2} - 1)^{2} : u^{6} - 2 u^{4} + u^{2}

= \int u^{6} - 2 u^{4} + u^{2} 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^{6} d u - \int 2 u^{4} d u + \int u^{2} d u

\int u^{6} d u = \frac{u ^{7}}{7}

\int 2 u^{4} d u = \frac{2 u ^{5}}{5}

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

= \frac{u ^{7}}{7} - \frac{2 u ^{5}}{5} + \frac{u ^{3}}{3}

代用を戻す

u = sec (x)

= \frac{sec ^{7} ( x )}{7} - \frac{2 sec ^{5} ( x )}{5} + \frac{sec ^{3} ( x )}{3}

定数を解答に追加する

= \frac{sec ^{7} ( x )}{7} - \frac{2 sec ^{5} ( x )}{5} + \frac{sec ^{3} ( x )}{3} + C

グラフ

人気の例

integral of x^2-2x-24

\int x^{2} - 2 x - 24 d x

integral of 6x^5+7

\int 6 x^{5} + 7 d x

integral of sin(k)x

\int sin (k) x d x

integral of ln(x)(x)

\int ln (x) (x) d x

integral of x^3cos(xy)

\int x^{3} cos (x y) d y