What is the integral of x^2sin(x)cos(x) ?

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

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integral of x^2sin(x)cos(x)

\int x^{2} sin (x) cos (x) d x

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

Solution steps

\int x^{2} sin (x) cos (x) d x

Rewrite using trig identities

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{1}{y ^{4}} d y

x \to 2 lim (+ (x))

\int \frac{2}{3} x^{\frac{2}{3}} d x

\int cos (5 x + 3) d x

f (x) = 10 (1 + 3 x)^{9}, at x = (0)

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