integral of xsin(1/x)

解答

\int x sin (\frac{1}{x}) d x

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

求解步骤

\int x sin (\frac{1}{x}) d x

使用换元积分法

= \int - \frac{sin ( u )}{u ^{3}} d u

提出常数:

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

= - \int \frac{sin ( u )}{u ^{3}} d u

使用分布积分法

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

\int - \frac{cos ( u )}{2 u ^{2}} d u = - \frac{1}{2} (- \frac{cos ( u )}{u} - Si (u))

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

u = \frac{1}{x}

代回

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

化简

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

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

解答补常数

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