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人気のある微分積分 >

integral of (sin^5(x))/(cos^7(x))

解

\int \frac{sin ^{5} ( x )}{cos ^{7} ( x )} d x

解

\frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x) + C

解答ステップ

\int \frac{sin ^{5} ( x )}{cos ^{7} ( x )} d x

三角関数の公式を使用して書き換える

= \int \frac{( 1 - cos ^{2} ( x ) ) ^{2} sin ( x )}{cos ^{7} ( x )} d x

u置換積分法を適用する

= \int - \frac{( 1 - u ^{2} ) ^{2}}{u ^{7}} d u

拡張

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

= \int - \frac{1}{u ^{7}} + \frac{2}{u ^{5}} - \frac{1}{u ^{3}} 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 \frac{1}{u ^{7}} d u + \int \frac{2}{u ^{5}} d u - \int \frac{1}{u ^{3}} d u

\int \frac{1}{u ^{7}} d u = - \frac{1}{6 u ^{6}}

\int \frac{2}{u ^{5}} d u = - \frac{1}{2 u ^{4}}

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

= - (- \frac{1}{6 u ^{6}}) - \frac{1}{2 u ^{4}} - (- \frac{1}{2 u ^{2}})

代用を戻す

u = cos (x)

= - (- \frac{1}{6 cos ^{6} ( x )}) - \frac{1}{2 cos ^{4} ( x )} - (- \frac{1}{2 cos ^{2} ( x )})

簡素化

- (- \frac{1}{6 cos ^{6} ( x )}) - \frac{1}{2 cos ^{4} ( x )} - (- \frac{1}{2 cos ^{2} ( x )}) : \frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x)

= \frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x)

定数を解答に追加する

= \frac{1}{6} sec^{6} (x) - \frac{1}{2} sec^{4} (x) + \frac{1}{2} sec^{2} (x) + C

グラフ

人気の例

integral of (2t-1)/(t^2+t)

\int \frac{2 t - 1}{t ^{2} + t} d t

integral of 2x^2sin(2x)

\int 2 x^{2} sin (2 x) d x

integral of e^xcos(npix)

\int e^{x} cos (nπ x) d x

integral of (e^{5x})/(e^{5x)-8}

\int \frac{e ^{5 x}}{e ^{5 x} - 8} d x

integral of 4^{2x+1}

\int 4^{2 x + 1} d x