What is the integral of x^3*cos(4x) ?

The integral of x^3*cos(4x) is 1/4 x^3sin(4x)-3/4 (-1/4 x^2cos(4x)+1/32 (4xsin(4x)+cos(4x)))+C

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integral of x^3*cos(4x)

\int x^{3} \cdot cos (4 x) d x

\frac{1}{4} x^{3} sin (4 x) - \frac{3}{4} (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))) + C

Solution steps

\int x^{3} cos (4 x) d x

Apply Integration By Parts

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

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

\int \frac{3}{4} x^{2} sin (4 x) d x = \frac{3}{4} (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)))

= \frac{1}{4} x^{3} sin (4 x) - \frac{3}{4} (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x)))

Add a constant to the solution

= \frac{1}{4} x^{3} sin (4 x) - \frac{3}{4} (- \frac{1}{4} x^{2} cos (4 x) + \frac{1}{32} (4 x sin (4 x) + cos (4 x))) + C

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What is the integral of x^3*cos(4x) ?
The integral of x^3*cos(4x) is 1/4 x^3sin(4x)-3/4 (-1/4 x^2cos(4x)+1/32 (4xsin(4x)+cos(4x)))+C