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

The integral of-4x^3cos(-4x) is -x^3sin(4x)-3/4 x^2cos(4x)+3/8 xsin(4x)+3/32 cos(4x)+C

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integral of-4x^3cos(-4x)

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

- x^{3} sin (4 x) - \frac{3}{4} x^{2} cos (4 x) + \frac{3}{8} x sin (4 x) + \frac{3}{32} cos (4 x) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Simplify

x^{3} cos (- 4 x) : x^{3} cos (4 x)

= - 4 \cdot \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

= - 4 (\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)))

= - 4 (\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))))

Simplify

- 4 (\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)))) : - x^{3} sin (4 x) - \frac{3}{4} x^{2} cos (4 x) + \frac{3}{8} x sin (4 x) + \frac{3}{32} cos (4 x)

= - x^{3} sin (4 x) - \frac{3}{4} x^{2} cos (4 x) + \frac{3}{8} x sin (4 x) + \frac{3}{32} cos (4 x)

Add a constant to the solution

= - x^{3} sin (4 x) - \frac{3}{4} x^{2} cos (4 x) + \frac{3}{8} x sin (4 x) + \frac{3}{32} cos (4 x) + C

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What is the integral of-4x^3cos(-4x) ?
The integral of-4x^3cos(-4x) is -x^3sin(4x)-3/4 x^2cos(4x)+3/8 xsin(4x)+3/32 cos(4x)+C