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人気のある微分積分 >

integral of (tan^5(x))/(sec^8(x))

解

\int \frac{tan ^{5} ( x )}{sec ^{8} ( x )} d x

解

- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

解答ステップ

\int \frac{tan ^{5} ( x )}{sec ^{8} ( x )} d x

三角関数の公式を使用して書き換える

= \int (1 - cos^{2} (x))^{2} sin (x) cos^{3} (x) d x

u置換積分法を適用する

= \int - u^{3} (1 - u^{2})^{2} d u

拡張

- u^{3} (1 - u^{2})^{2} : - u^{3} + 2 u^{5} - u^{7}

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

\int u^{3} d u = \frac{u ^{4}}{4}

\int 2 u^{5} d u = \frac{u ^{6}}{3}

\int u^{7} d u = \frac{u ^{8}}{8}

= - \frac{u ^{4}}{4} + \frac{u ^{6}}{3} - \frac{u ^{8}}{8}

代用を戻す

u = cos (x)

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8}

定数を解答に追加する

= - \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{3} - \frac{cos ^{8} ( x )}{8} + C

グラフ

人気の例

integral from 0 to 4 of pi(16x^2-x^4)

\int_{0}^{4} π (16 x^{2} - x^{4}) d x

integral of (2x)/(1-8x^2)

\int \frac{2 x}{1 - 8 x ^{2}} d x

integral of (e^{7sqrt(t)})/(sqrt(t))

\int \frac{e ^{7 t}}{t} d t

limit as x approaches 5 of (3x^2-13x-10)/(2x^2-7x-15)

x \to 5 lim (\frac{3 x ^{2} - 13 x - 10}{2 x ^{2} - 7 x - 15})

derivative of (x^2-3x/(sqrt(x+1)))

\frac{d}{d x} (\frac{x ^{2} - 3 x}{x + 1})