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人気のある微分積分 >

integral of cos^3(xsi)n^2x

解

\int cos^{3} (x s i) n^{2} x d x

解

n^{2} \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^{2} x d x

定数を除く:

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

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

u置換積分法を適用する

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

部分積分を適用する

= n^{2} \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^{2} \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^{2} \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^{2} \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^{2} \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

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