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受欢迎的微积分 >

integral of 4/(x^5)cos(1/(x^2))

解答

\int \frac{4}{x ^{5}} cos (\frac{1}{x ^{2}}) d x

解答

- 2 (\frac{sin ( \frac{1}{x ^{2}} )}{x ^{2}} + cos (\frac{1}{x ^{2}})) + C

求解步骤

\int \frac{4}{x ^{5}} cos (\frac{1}{x ^{2}}) d x

提出常数:

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

= 4 \cdot \int \frac{1}{x ^{5}} cos (\frac{1}{x ^{2}}) d x

使用换元积分法

= 4 \cdot \int - \frac{u cos ( u )}{2} d u

提出常数:

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

= 4 (- \frac{1}{2} \cdot \int u cos (u) d u)

使用分布积分法

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

使用积分常见定则:

\int sin (u) d u = - cos (u)

= 4 (- \frac{1}{2} (u sin (u) - (- cos (u))))

u = \frac{1}{x ^{2}}

代回

= 4 (- \frac{1}{2} (\frac{1}{x ^{2}} sin (\frac{1}{x ^{2}}) - (- cos (\frac{1}{x ^{2}}))))

化简

4 (- \frac{1}{2} (\frac{1}{x ^{2}} sin (\frac{1}{x ^{2}}) - (- cos (\frac{1}{x ^{2}})))) : - 2 (\frac{sin ( \frac{1}{x ^{2}} )}{x ^{2}} + cos (\frac{1}{x ^{2}}))

= - 2 (\frac{sin ( \frac{1}{x ^{2}} )}{x ^{2}} + cos (\frac{1}{x ^{2}}))

解答补常数

= - 2 (\frac{sin ( \frac{1}{x ^{2}} )}{x ^{2}} + cos (\frac{1}{x ^{2}})) + C

作图

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