What is the integral of x(sin(x)-1) ?

The integral of x(sin(x)-1) is -xcos(x)+sin(x)-(x^2)/2+C

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integral of x(sin(x)-1)

\int x (sin (x) - 1) d x

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

Solution steps

\int x (sin (x) - 1) d x

Expand

x (sin (x) - 1) : x sin (x) - x

= \int x sin (x) - 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 d x

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

\int x d x = \frac{x ^{2}}{2}

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

Add a constant to the solution

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

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What is the integral of x(sin(x)-1) ?
The integral of x(sin(x)-1) is -xcos(x)+sin(x)-(x^2)/2+C