What is the integral of sin(x)cos(x)x ?

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

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integral of sin(x)cos(x)x

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

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

Solution steps

\int sin (x) cos (x) 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

x \to 1 + lim (\frac{5 x + 6}{x - 1})

x = - 5, x = 1, y = 7 x, y = x^{2} - 8

t^{3} + 1 t, at t = 1

\frac{\partial}{\partial u} (3 y uxu v^{3} - 2)

\int \frac{1}{100 - 2 x} d x

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