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人気のある微分積分 >

integral of (sin^4(cot(x)))/(sin^2(x))

解

\int \frac{sin ^{4} ( cot ( x ) )}{sin ^{2} ( x )} d x

解

\frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x)) + C

解答ステップ

\int \frac{sin ^{4} ( cot ( x ) )}{sin ^{2} ( x )} d x

u置換積分法を適用する

= \int - sin^{4} (u) d u

定数を除く:

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

= - \int sin^{4} (u) d u

積分換算を適用する

= - (- \frac{cos ( u ) sin ^{3} ( u )}{4} + \frac{3}{4} \cdot \int sin^{2} (u) d u)

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

= - (- \frac{cos ( u ) sin ^{3} ( u )}{4} + \frac{3}{4} \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u)))

代用を戻す

u = cot (x)

= - (- \frac{cos ( cot ( x ) ) sin ^{3} ( cot ( x ) )}{4} + \frac{3}{4} \cdot \frac{1}{2} (cot (x) - \frac{1}{2} sin (2 cot (x))))

簡素化

- (- \frac{cos ( cot ( x ) ) sin ^{3} ( cot ( x ) )}{4} + \frac{3}{4} \cdot \frac{1}{2} (cot (x) - \frac{1}{2} sin (2 cot (x)))) : \frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x))

= \frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x))

定数を解答に追加する

= \frac{1}{4} sin^{3} (cot (x)) cos (cot (x)) - \frac{3}{8} cot (x) + \frac{3}{16} sin (2 cot (x)) + C

グラフ

人気の例

(\partial)/(\partial y)(tan(x^3y^2))

\frac{\partial}{\partial y} (tan (x^{3} y^{2}))

(\partial)/(\partial x)(cos(x^2-y))

\frac{\partial}{\partial x} (cos (x^{2} - y))

integral of (2x)/(sqrt(x-1))

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

(\partial)/(\partial y)(3y)

\frac{\partial}{\partial y} (3 y)

4y^{''}-4y^'+4y=0

4 y^{^{''}} - 4 y^{^{'}} + 4 y = 0