What is the integral of x^4cos(2x) ?

The integral of x^4cos(2x) is 1/2 x^4sin(2x)+x^3cos(2x)-3/2 x^2sin(2x)-3/2 xcos(2x)+3/4 sin(2x)+C

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integral of x^4cos(2x)

\int x^{4} cos (2 x) d x

\frac{1}{2} x^{4} sin (2 x) + x^{3} cos (2 x) - \frac{3}{2} x^{2} sin (2 x) - \frac{3}{2} x cos (2 x) + \frac{3}{4} sin (2 x) + C

Solution steps

\int x^{4} cos (2 x) d x

Apply Integration By Parts

: \frac{1}{2} x^{4} sin (2 x) - \int 2 x^{3} sin (2 x) d x

= \frac{1}{2} x^{4} sin (2 x) - \int 2 x^{3} sin (2 x) d x

\int 2 x^{3} sin (2 x) d x = - x^{3} cos (2 x) + \frac{3}{2} x^{2} sin (2 x) + \frac{3}{2} x cos (2 x) - \frac{3}{4} sin (2 x)

= \frac{1}{2} x^{4} sin (2 x) - (- x^{3} cos (2 x) + \frac{3}{2} x^{2} sin (2 x) + \frac{3}{2} x cos (2 x) - \frac{3}{4} sin (2 x))

Simplify

= \frac{1}{2} x^{4} sin (2 x) + x^{3} cos (2 x) - \frac{3}{2} x^{2} sin (2 x) - \frac{3}{2} x cos (2 x) + \frac{3}{4} sin (2 x)

Add a constant to the solution

= \frac{1}{2} x^{4} sin (2 x) + x^{3} cos (2 x) - \frac{3}{2} x^{2} sin (2 x) - \frac{3}{2} x cos (2 x) + \frac{3}{4} sin (2 x) + C

9 - x^{2}, y = 0

\frac{d}{d x} (x sin (6 x))

\int 16 y^{3} + 9 y^{2} - 6 y + 3 d y

\int_{0}^{1} 4^{4 x} d x

What is the integral of x^4cos(2x) ?
The integral of x^4cos(2x) is 1/2 x^4sin(2x)+x^3cos(2x)-3/2 x^2sin(2x)-3/2 xcos(2x)+3/4 sin(2x)+C