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人気のある微分積分 >

integral of cos^3(6xsi)n^{-2}6x

解

\int cos^{3} (6 x s i) n^{- 2} 6 x

解

\frac{1}{54 s ^{2} i ^{2} n ^{2}} (54 s i x sin (6 s i x) - 18 s i x sin^{3} (6 s i x) + cos^{3} (6 s i x) + 6 cos (6 s i x)) + C

解答ステップ

\int cos^{3} (6 x s i) n^{- 2} \cdot 6 x d x

定数を除く:

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

= 6 n^{- 2} \cdot \int cos^{3} (6 s i x) x d x

簡素化

6 n^{- 2} : \frac{6}{n ^{2}}

= \frac{6}{n ^{2}} \cdot \int cos^{3} (6 s i x) x d x

u置換積分法を適用する

= \frac{6}{n ^{2}} \cdot \int \frac{u cos ^{3} ( u )}{36 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

= \frac{6}{n ^{2}} \cdot \frac{1}{36 s ^{2} i ^{2}} \cdot \int u cos^{3} (u) d u

部分積分を適用する

= \frac{6}{n ^{2}} \cdot \frac{1}{36 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)

= \frac{6}{n ^{2}} \cdot \frac{1}{36 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 = 6 s i x

= \frac{6}{n ^{2}} \cdot \frac{1}{36 s ^{2} i ^{2}} (6 s i x (sin (6 s i x) - \frac{1}{3} sin^{3} (6 s i x)) - (- \frac{2}{3} cos (6 s i x) - \frac{1}{9} cos^{3} (6 s i x)))

簡素化

\frac{6}{n ^{2}} \cdot \frac{1}{36 s ^{2} i ^{2}} (6 s i x (sin (6 s i x) - \frac{1}{3} sin^{3} (6 s i x)) - (- \frac{2}{3} cos (6 s i x) - \frac{1}{9} cos^{3} (6 s i x))) : \frac{1}{54 s ^{2} i ^{2} n ^{2}} (54 s i x sin (6 s i x) - 18 s i x sin^{3} (6 s i x) + cos^{3} (6 s i x) + 6 cos (6 s i x))

= \frac{1}{54 s ^{2} i ^{2} n ^{2}} (54 s i x sin (6 s i x) - 18 s i x sin^{3} (6 s i x) + cos^{3} (6 s i x) + 6 cos (6 s i x))

定数を解答に追加する

= \frac{1}{54 s ^{2} i ^{2} n ^{2}} (54 s i x sin (6 s i x) - 18 s i x sin^{3} (6 s i x) + cos^{3} (6 s i x) + 6 cos (6 s i x)) + C

グラフ

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