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

x \to 0 lim (\frac{ln ( 1 + \frac{1}{x} )}{x})

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

\int (cos (x))^{5} sin (x) d x

\int (x^{2} + 2 x)^{5} (2 x + 2) d x

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

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