What is the integral of xcos^3(x) ?

The integral of xcos^3(x) is xsin(x)-1/3 xsin^3(x)+2/3 cos(x)+1/9 cos^3(x)+C

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integral of xcos^3(x)

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

x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x) + C

Solution steps

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

Apply Integration By Parts

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

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

\int sin (x) - \frac{1}{3} sin^{3} (x) d x = - \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)

= x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x))

Simplify

x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)) : x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)

= x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)

Add a constant to the solution

= x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x) + C

\frac{x ^{3}}{3} + 2 x^{2} + 4 x - 1

x \to 0 lim (6 sin (x) - 1)

\frac{lo g _{10} ( 2 - x )}{x + 3}

\frac{d}{d x} ((4 x - 5)^{2})

n = 5 \sum \infty \frac{1}{n ^{2} - 1}

What is the integral of xcos^3(x) ?
The integral of xcos^3(x) is xsin(x)-1/3 xsin^3(x)+2/3 cos(x)+1/9 cos^3(x)+C