What is the integral of xsin(1/x) ?

The integral of xsin(1/x) is 1/2 x^2sin(1/x)+1/2 xcos(1/x)+1/2 \Si(1/x)+C

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integral of xsin(1/x)

\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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

= - (- \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))))

Substitute back

u = \frac{1}{x}

= - (- \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}))))

Simplify

- (- \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})

Add a constant to the solution

= \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

\int \frac{2}{2 x} d x

\frac{d}{d t} (2 cos (t) - 2)

\int \frac{1}{1 + x + x ^{2}} d x

\int y^{- 3} d y

\frac{\partial}{\partial y} (x y^{2} arctan (z))

What is the integral of xsin(1/x) ?
The integral of xsin(1/x) is 1/2 x^2sin(1/x)+1/2 xcos(1/x)+1/2 \Si(1/x)+C