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

The integral of x(sin(x)-cos(x)) is -xcos(x)+sin(x)-xsin(x)-cos(x)+C

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integral of x(sin(x)-cos(x))

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

- x cos (x) + sin (x) - x sin (x) - cos (x) + C

Solution steps

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

Expand

x (sin (x) - cos (x)) : x sin (x) - x cos (x)

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

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

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

= - x cos (x) + sin (x) - (x sin (x) + cos (x))

Simplify

= - x cos (x) + sin (x) - x sin (x) - cos (x)

Add a constant to the solution

= - x cos (x) + sin (x) - x sin (x) - cos (x) + C

\int (3 x + 9)^{6} d x

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\int 24 x + 24 x^{2} d x

\int x^{\frac{1}{2}} + 3 d x

\int \frac{5 x + 6}{x ^{2} + x + 1} d x

What is the integral of x(sin(x)-cos(x)) ?
The integral of x(sin(x)-cos(x)) is -xcos(x)+sin(x)-xsin(x)-cos(x)+C