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人気のある微分積分 >

integral of 70sin^4(x)cos^3(x)

解

\int 70 sin^{4} (x) cos^{3} (x) d x

解

70 (\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}) + C

解答ステップ

\int 70 sin^{4} (x) cos^{3} (x) d x

定数を除く:

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

= 70 \cdot \int sin^{4} (x) cos^{3} (x) d x

三角関数の公式を使用して書き換える

= 70 \cdot \int (1 - sin^{2} (x)) cos (x) sin^{4} (x) d x

u置換積分法を適用する

= 70 \cdot \int u^{4} (1 - u^{2}) d u

拡張

u^{4} (1 - u^{2}) : u^{4} - u^{6}

= 70 \cdot \int u^{4} - u^{6} 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

= 70 (\int u^{4} d u - \int u^{6} d u)

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

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

= 70 (\frac{u ^{5}}{5} - \frac{u ^{7}}{7})

代用を戻す

u = sin (x)

= 70 (\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7})

定数を解答に追加する

= 70 (\frac{sin ^{5} ( x )}{5} - \frac{sin ^{7} ( x )}{7}) + C

グラフ

人気の例

integral from 1 to 2 of 4x^2

\int_{1}^{2} 4 x^{2} d x

limit as x approaches 5-of f(x)

x \to 5 - lim (f (x))

limit as x approaches 0 of (ln(x))/(x+1)

x \to 0 lim (\frac{ln ( x )}{x + 1})

tangent f(x)=(7x)/(x-3),\at x=6

t an g e n t f (x) = \frac{7 x}{x - 3}, at x = 6

derivative 2+5tan^{-1}(x/2)

d er i v a t i v e 2 + 5 tan^{- 1} (\frac{x}{2})