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人気のある微分積分 >

integral of (sin(x))^6(cos(x))^3

解

\int (sin (x))^{6} (cos (x))^{3} d x

解

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

解答ステップ

\int (sin (x))^{6} (cos (x))^{3} d x

簡素化

= \int sin^{6} (x) cos^{3} (x) d x

三角関数の公式を使用して書き換える

= \int (1 - sin^{2} (x)) cos (x) sin^{6} (x) d x

u置換積分法を適用する

= \int u^{6} (1 - u^{2}) d u

拡張

u^{6} (1 - u^{2}) : u^{6} - u^{8}

= \int u^{6} - u^{8} 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^{6} d u - \int u^{8} d u

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

\int u^{8} d u = \frac{u ^{9}}{9}

= \frac{u ^{7}}{7} - \frac{u ^{9}}{9}

代用を戻す

u = sin (x)

= \frac{sin ^{7} ( x )}{7} - \frac{sin ^{9} ( x )}{9}

定数を解答に追加する

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

グラフ

人気の例

integral of sec^6(4x)

\int sec^{6} (4 x) d x

tangent f(x)=(3x^3-4x^2+5x-9)^2,\at x=2

t an g e n t f (x) = (3 x^{3} - 4 x^{2} + 5 x - 9)^{2}, at x = 2

limit as x approaches 2 of x^2+6x+5

x \to 2 lim (x^{2} + 6 x + 5)

limit as x approaches 0+of x^{13x}

x \to 0 + lim (x^{13 x})

limit as x approaches 4 of 6

x \to 4 lim (6)