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人気のある微分積分 >

integral of-sin^3(x)cos^2(x)

解

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

解

\frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x) + C

解答ステップ

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

定数を除く:

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

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

u置換積分法を適用する

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

拡張

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

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

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

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

= - (- \frac{u ^{3}}{3} + \frac{u ^{5}}{5})

代用を戻す

u = cos (x)

= - (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5})

簡素化

- (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5}) : \frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x)

= \frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x)

定数を解答に追加する

= \frac{1}{3} cos^{3} (x) - \frac{1}{5} cos^{5} (x) + C

グラフ

人気の例

integral of 1/(x(x^2+x+5))

\int \frac{1}{x ( x ^{2} + x + 5 )} d x

(\partial)/(\partial x)((6-x^2y^2)^2)

\frac{\partial}{\partial x} ((6 - x^{2} y^{2})^{2})

limit as x approaches 4+of (4+x)/(4-x)

x \to 4 + lim (\frac{4 + x}{4 - x})

limit as h approaches 0 of 6x+1+3h

h \to 0 lim (6 x + 1 + 3 h)

inverselaplace {1/(s^2+s-20)}

in v erse l a pl a ce {\frac{1}{s ^{2} + s - 20}}