What is the integral of x^3cos(2x) ?

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

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integral of x^3cos(2x)

\int x^{3} cos (2 x) d x

\frac{1}{2} x^{3} sin (2 x) - \frac{3}{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^{3} cos (2 x) d x

Apply Integration By Parts

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

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

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

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

\frac{e ^{x} + e ^{- x}}{2}

\int (6 e^{5 x} - \frac{7}{x}) d x

\frac{d}{d x} (\frac{6 x - 3}{x - 3})

\int sin (6 t) d t

\frac{d}{d x} (\frac{sin ( 0.5 x )}{x})

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