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受欢迎的微积分 >

integral of cos^5(xta)n^3x

解答

\int cos^{5} (x t a) n^{3} x d x

解答

\frac{n ^{3}}{225 a ^{2} t ^{2}} (225 a t x sin (a t x) - 150 a t x sin^{3} (a t x) + 45 a t x sin^{5} (a t x) + 120 cos (a t x) + 20 cos^{3} (a t x) + 9 cos^{5} (a t x)) + C

求解步骤

\int cos^{5} (x t a) n^{3} x d x

提出常数:

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

= n^{3} \cdot \int cos^{5} (a t x) x d x

使用换元积分法

= n^{3} \cdot \int \frac{u cos ^{5} ( u )}{a ^{2} t ^{2}} d u

提出常数:

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

= n^{3} \frac{1}{a ^{2} t ^{2}} \cdot \int u cos^{5} (u) d u

使用分布积分法

= n^{3} \frac{1}{a ^{2} t ^{2}} (u (sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u)) - \int sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u) d u)

\int sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u) d u = - \frac{8}{15} cos (u) - \frac{2}{9} cos^{3} (u) + \frac{2}{15} cos^{3} (u) - \frac{1}{25} cos^{5} (u)

= n^{3} \frac{1}{a ^{2} t ^{2}} (u (sin (u) - \frac{2}{3} sin^{3} (u) + \frac{1}{5} sin^{5} (u)) - (- \frac{8}{15} cos (u) - \frac{2}{9} cos^{3} (u) + \frac{2}{15} cos^{3} (u) - \frac{1}{25} cos^{5} (u)))

u = a t x

代回

= n^{3} \frac{1}{a ^{2} t ^{2}} (a t x (sin (a t x) - \frac{2}{3} sin^{3} (a t x) + \frac{1}{5} sin^{5} (a t x)) - (- \frac{8}{15} cos (a t x) - \frac{2}{9} cos^{3} (a t x) + \frac{2}{15} cos^{3} (a t x) - \frac{1}{25} cos^{5} (a t x)))

化简

n^{3} \frac{1}{a ^{2} t ^{2}} (a t x (sin (a t x) - \frac{2}{3} sin^{3} (a t x) + \frac{1}{5} sin^{5} (a t x)) - (- \frac{8}{15} cos (a t x) - \frac{2}{9} cos^{3} (a t x) + \frac{2}{15} cos^{3} (a t x) - \frac{1}{25} cos^{5} (a t x))) : \frac{n ^{3}}{225 a ^{2} t ^{2}} (225 a t x sin (a t x) - 150 a t x sin^{3} (a t x) + 45 a t x sin^{5} (a t x) + 120 cos (a t x) + 20 cos^{3} (a t x) + 9 cos^{5} (a t x))

= \frac{n ^{3}}{225 a ^{2} t ^{2}} (225 a t x sin (a t x) - 150 a t x sin^{3} (a t x) + 45 a t x sin^{5} (a t x) + 120 cos (a t x) + 20 cos^{3} (a t x) + 9 cos^{5} (a t x))

解答补常数

= \frac{n ^{3}}{225 a ^{2} t ^{2}} (225 a t x sin (a t x) - 150 a t x sin^{3} (a t x) + 45 a t x sin^{5} (a t x) + 120 cos (a t x) + 20 cos^{3} (a t x) + 9 cos^{5} (a t x)) + C

作图

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