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人気のある微分積分 >

integral of cos^3(xsi)n^4x

解

\int cos^{3} (x s i) n^{4} x d x

解

n^{4} \frac{1}{s ^{2} i ^{2}} (s i x sin (s i x) - \frac{1}{3} s i x sin^{3} (s i x) + \frac{2}{3} cos (s i x) + \frac{1}{9} cos^{3} (s i x)) + C

解答ステップ

\int cos^{3} (x s i) n^{4} x d x

定数を除く:

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

= n^{4} \cdot \int cos^{3} (s i x) x d x

u置換積分法を適用する

= n^{4} \cdot \int \frac{u cos ^{3} ( u )}{s ^{2} i ^{2}} d u

定数を除く:

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

= n^{4} \frac{1}{s ^{2} i ^{2}} \cdot \int u cos^{3} (u) d u

部分積分を適用する

= n^{4} \frac{1}{s ^{2} i ^{2}} (u (sin (u) - \frac{1}{3} sin^{3} (u)) - \int sin (u) - \frac{1}{3} sin^{3} (u) d u)

\int sin (u) - \frac{1}{3} sin^{3} (u) d u = - \frac{2}{3} cos (u) - \frac{1}{9} cos^{3} (u)

= n^{4} \frac{1}{s ^{2} i ^{2}} (u (sin (u) - \frac{1}{3} sin^{3} (u)) - (- \frac{2}{3} cos (u) - \frac{1}{9} cos^{3} (u)))

代用を戻す

u = s i x

= n^{4} \frac{1}{s ^{2} i ^{2}} (s i x (sin (s i x) - \frac{1}{3} sin^{3} (s i x)) - (- \frac{2}{3} cos (s i x) - \frac{1}{9} cos^{3} (s i x)))

拡張

s i x (sin (s i x) - \frac{1}{3} sin^{3} (s i x)) - (- \frac{2}{3} cos (s i x) - \frac{1}{9} cos^{3} (s i x)) : s i x sin (s i x) - \frac{1}{3} s i x sin^{3} (s i x) + \frac{2}{3} cos (s i x) + \frac{1}{9} cos^{3} (s i x)

= n^{4} \frac{1}{s ^{2} i ^{2}} (s i x sin (s i x) - \frac{1}{3} s i x sin^{3} (s i x) + \frac{2}{3} cos (s i x) + \frac{1}{9} cos^{3} (s i x))

定数を解答に追加する

= n^{4} \frac{1}{s ^{2} i ^{2}} (s i x sin (s i x) - \frac{1}{3} s i x sin^{3} (s i x) + \frac{2}{3} cos (s i x) + \frac{1}{9} cos^{3} (s i x)) + C

グラフ

人気の例

integral of (x^3)/(324)

\int \frac{x ^{3}}{324} d x

integral from 0 to infinity of x/(1+x^3)

\int_{0}^{\infty} \frac{x}{1 + x ^{3}} d x

limit as x approaches 4 of x^2

x \to 4 lim (x^{2})

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

y^{''}+4y=x^2+3e^x,y(0)=0,y^'(0)=2

y^{^{''}} + 4 y = x^{2} + 3 e^{x}, y (0) = 0, y^{^{'}} (0) = 2