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受欢迎的微积分 >

integral of (cos(pi/(x^{15)}))/(x^{16)}

解答

\int \frac{cos ( \frac{π}{x ^{15}} )}{x ^{16}} d x

解答

- \frac{1}{15 π} sin (\frac{π}{x ^{15}}) + C

求解步骤

\int \frac{cos ( \frac{π}{x ^{15}} )}{x ^{16}} d x

使用换元积分法

= \int \frac{cos ( \frac{π}{u} )}{15 u ^{2}} d u

提出常数:

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

= \frac{1}{15} \cdot \int \frac{cos ( \frac{π}{u} )}{u ^{2}} d u

使用换元积分法

= \frac{1}{15} \cdot \int - \frac{cos ( v )}{π} d v

提出常数:

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

= \frac{1}{15} (- \frac{1}{π} \cdot \int cos (v) d v)

使用积分常见定则:

\int cos (v) d v = sin (v)

= \frac{1}{15} (- \frac{1}{π} sin (v))

代回

= \frac{1}{15} (- \frac{1}{π} sin (\frac{π}{x ^{15}}))

化简

\frac{1}{15} (- \frac{1}{π} sin (\frac{π}{x ^{15}})) : - \frac{1}{15 π} sin (\frac{π}{x ^{15}})

= - \frac{1}{15 π} sin (\frac{π}{x ^{15}})

解答补常数

= - \frac{1}{15 π} sin (\frac{π}{x ^{15}}) + C

作图

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