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

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

\frac{d}{d x} ((2 - x)^{2})

\frac{\partial}{\partial x} (4 x y - 10 x^{2} y^{2} z + 5 z^{2})

y^{\frac{4}{5}}

x \to 1 - lim (\frac{2 x}{x ^{3} - 1})

\frac{d}{d x} ((x^{4} + 8)^{\frac{1}{2}})

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