What is the integral of x(xsin(x)-cos(x)) ?

The integral of x(xsin(x)-cos(x)) is -x^2cos(x)+xsin(x)+cos(x)+C

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integral of x(xsin(x)-cos(x))

\int x (x sin (x) - cos (x)) d x

- x^{2} cos (x) + x sin (x) + cos (x) + C

Solution steps

\int x (x sin (x) - cos (x)) d x

Expand

x (x sin (x) - cos (x)) : x^{2} sin (x) - x cos (x)

= \int x^{2} sin (x) - x cos (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} sin (x) d x - \int x cos (x) d x

\int x^{2} sin (x) d x = - x^{2} cos (x) + 2 (x sin (x) + cos (x))

\int x cos (x) d x = x sin (x) + cos (x)

= - x^{2} cos (x) + 2 (x sin (x) + cos (x)) - (x sin (x) + cos (x))

Simplify

= - x^{2} cos (x) + x sin (x) + cos (x)

Add a constant to the solution

= - x^{2} cos (x) + x sin (x) + cos (x) + C

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y = 2 (x + y) y^{^{'}}

\int 3 sin^{6} (x) cos (x) d x

x \to 0 lim (100)

What is the integral of x(xsin(x)-cos(x)) ?
The integral of x(xsin(x)-cos(x)) is -x^2cos(x)+xsin(x)+cos(x)+C