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人気のある微分積分 >

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

解

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

解

7 (2 sin (x) - \frac{4}{5} sin^{\frac{5}{2}} (x) + \frac{2}{9} sin^{\frac{9}{2}} (x)) + C

解答ステップ

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

定数を除く:

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

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

三角関数の公式を使用して書き換える

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

u置換積分法を適用する

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

拡張

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

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

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

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

\int 2 u^{\frac{3}{2}} d u = \frac{4}{5} u^{\frac{5}{2}}

\int u^{\frac{7}{2}} d u = \frac{2}{9} u^{\frac{9}{2}}

= 7 (2 u - \frac{4}{5} u^{\frac{5}{2}} + \frac{2}{9} u^{\frac{9}{2}})

代用を戻す

u = sin (x)

= 7 (2 sin (x) - \frac{4}{5} sin^{\frac{5}{2}} (x) + \frac{2}{9} sin^{\frac{9}{2}} (x))

定数を解答に追加する

= 7 (2 sin (x) - \frac{4}{5} sin^{\frac{5}{2}} (x) + \frac{2}{9} sin^{\frac{9}{2}} (x)) + C

グラフ

人気の例

integral of 5xln(6x)

\int 5 x ln (6 x) d x

derivative y=x^3+4

d er i v a t i v e y = x^{3} + 4

derivative of cos^2(sin(5e^x-16^x))

\frac{d}{d x} (cos^{2} (sin (5 e^{x} - 1 6^{x})))

integral of 1/(sqrt(t^3))

\int \frac{1}{t ^{3}} d t

(\partial)/(\partial x)((x)cos(x)cos(y))

\frac{\partial}{\partial x} ((x) cos (x) cos (y))