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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

提出常数:

\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

使用换元积分法

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

提出常数:

\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

使用分布积分法

= 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)))

u = s i x

代回

= 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)))

乘开

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))

解答补常数

= 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

作图

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