What is the integral of-xsin(x)cos(x) ?

The integral of-xsin(x)cos(x) is -1/2 (-1/2 xcos(2x)+1/4 sin(2x))+C

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integral of-xsin(x)cos(x)

\int - x sin (x) cos (x) d x

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

Solution steps

\int - x sin (x) cos (x) d x

Take the constant out:

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

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

Rewrite using trig identities

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

Take the constant out:

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

= - \frac{1}{2} \cdot \int x sin (2 x) d x

Apply Integration By Parts

: - \frac{1}{2} x cos (2 x) - \int - \frac{1}{2} cos (2 x) d x

= - \frac{1}{2} (- \frac{1}{2} x cos (2 x) - \int - \frac{1}{2} cos (2 x) d x)

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{1}{3} sin (x - 3) + e^{x + 3} + x^{- 3} d x

x \to 0 lim ((2 + x)^{3} - 8)

\int \frac{2 x - 1}{x ^{2} + 4 x + 5} d x

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

\int x \cdot e^{\frac{x ^{2}}{2}} d x

What is the integral of-xsin(x)cos(x) ?
The integral of-xsin(x)cos(x) is -1/2 (-1/2 xcos(2x)+1/4 sin(2x))+C