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人気のある微分積分 >

integral of 3sin^3(xco)s^7x

解

\int 3 sin^{3} (x co) s^{7} x d x

解

- \frac{3 s ^{7}}{co} x cos (co x) + \frac{s ^{7}}{co} x cos^{3} (co x) + \frac{2 s ^{7}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{7}}{3 c ^{2} o ^{2}} sin^{3} (co x) + C

解答ステップ

\int 3 sin^{3} (x co) s^{7} x d x

定数を除く:

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

= 3 s^{7} \cdot \int sin^{3} (co x) x d x

u置換積分法を適用する

= 3 s^{7} \cdot \int \frac{u sin ^{3} ( u )}{c ^{2} o ^{2}} d u

定数を除く:

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

= 3 s^{7} \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

部分積分を適用する

= 3 s^{7} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - \int - cos (u) + \frac{1}{3} cos^{3} (u) d u)

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

= 3 s^{7} \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - (- \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)))

代用を戻す

u = co x

= 3 s^{7} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x)))

簡素化

3 s^{7} \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x))) : - \frac{3 s ^{7}}{co} x cos (co x) + \frac{s ^{7}}{co} x cos^{3} (co x) + \frac{2 s ^{7}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{7}}{3 c ^{2} o ^{2}} sin^{3} (co x)

= - \frac{3 s ^{7}}{co} x cos (co x) + \frac{s ^{7}}{co} x cos^{3} (co x) + \frac{2 s ^{7}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{7}}{3 c ^{2} o ^{2}} sin^{3} (co x)

定数を解答に追加する

= - \frac{3 s ^{7}}{co} x cos (co x) + \frac{s ^{7}}{co} x cos^{3} (co x) + \frac{2 s ^{7}}{c ^{2} o ^{2}} sin (co x) + \frac{s ^{7}}{3 c ^{2} o ^{2}} sin^{3} (co x) + C

グラフ

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