What is the integral of (-4x+1)(sin(x)) ?

The integral of (-4x+1)(sin(x)) is -4(-xcos(x)+sin(x))-cos(x)+C

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integral of (-4x+1)(sin(x))

\int (- 4 x + 1) (sin (x)) d x

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

Solution steps

\int (- 4 x + 1) sin (x) d x

Expand

(- 4 x + 1) sin (x) : - 4 x sin (x) + sin (x)

= \int - 4 x sin (x) + sin (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 4 x sin (x) d x + \int sin (x) d x

\int 4 x sin (x) d x = 4 (- x cos (x) + sin (x))

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

= - 4 (- x cos (x) + sin (x)) - cos (x)

Add a constant to the solution

= - 4 (- x cos (x) + sin (x)) - cos (x) + C

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What is the integral of (-4x+1)(sin(x)) ?
The integral of (-4x+1)(sin(x)) is -4(-xcos(x)+sin(x))-cos(x)+C