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受欢迎的微积分 >

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

解答

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

解答

\frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x)) + C

求解步骤

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

使用换元积分法

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

使用分布积分法

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} u (u + \frac{1}{2} sin (2 u)) - \int \frac{1}{2} (u + \frac{1}{2} sin (2 u)) d u)

\int \frac{1}{2} (u + \frac{1}{2} sin (2 u)) d u = \frac{1}{2} (\frac{u ^{2}}{2} - \frac{1}{4} cos (2 u))

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} u (u + \frac{1}{2} sin (2 u)) - \frac{1}{2} (\frac{u ^{2}}{2} - \frac{1}{4} cos (2 u)))

u = s i x

代回

= n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} s i x (s i x + \frac{1}{2} sin (2 s i x)) - \frac{1}{2} (\frac{( s i x ) ^{2}}{2} - \frac{1}{4} cos (2 s i x)))

化简

n^{3} \frac{1}{s ^{2} i ^{2}} (\frac{1}{2} s i x (s i x + \frac{1}{2} sin (2 s i x)) - \frac{1}{2} (\frac{( s i x ) ^{2}}{2} - \frac{1}{4} cos (2 s i x))) : \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x))

= \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x))

解答补常数

= \frac{n ^{3}}{8 s ^{2} i ^{2}} (2 s^{2} i^{2} x^{2} + 2 s i x sin (2 s i x) + cos (2 s i x)) + C

作图

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