What is the integral of (ax+2)sin(x) ?

The integral of (ax+2)sin(x) is a(-xcos(x)+sin(x))-2cos(x)+C

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integral of (ax+2)sin(x)

\int (a x + 2) sin (x) d x

a (- x cos (x) + sin (x)) - 2 cos (x) + C

Solution steps

\int (a x + 2) sin (x) d x

Expand

(a x + 2) sin (x) : a x sin (x) + 2 sin (x)

= \int a x sin (x) + 2 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 a x sin (x) d x + \int 2 sin (x) d x

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

\int 2 sin (x) d x = - 2 cos (x)

= a (- x cos (x) + sin (x)) - 2 cos (x)

Add a constant to the solution

= a (- x cos (x) + sin (x)) - 2 cos (x) + C

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What is the integral of (ax+2)sin(x) ?
The integral of (ax+2)sin(x) is a(-xcos(x)+sin(x))-2cos(x)+C