What is the integral of (5x^2+3)cos(x) ?

The integral of (5x^2+3)cos(x) is 5x^2sin(x)+10xcos(x)-7sin(x)+C

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integral of (5x^2+3)cos(x)

\int (5 x^{2} + 3) cos (x) d x

5 x^{2} sin (x) + 10 x cos (x) - 7 sin (x) + C

Solution steps

\int (5 x^{2} + 3) cos (x) d x

Expand

(5 x^{2} + 3) cos (x) : 5 x^{2} cos (x) + 3 cos (x)

= \int 5 x^{2} cos (x) + 3 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 5 x^{2} cos (x) d x + \int 3 cos (x) d x

\int 5 x^{2} cos (x) d x = 5 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))

\int 3 cos (x) d x = 3 sin (x)

= 5 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) + 3 sin (x)

Simplify

5 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))) + 3 sin (x) : 5 x^{2} sin (x) + 10 x cos (x) - 7 sin (x)

= 5 x^{2} sin (x) + 10 x cos (x) - 7 sin (x)

Add a constant to the solution

= 5 x^{2} sin (x) + 10 x cos (x) - 7 sin (x) + C

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What is the integral of (5x^2+3)cos(x) ?
The integral of (5x^2+3)cos(x) is 5x^2sin(x)+10xcos(x)-7sin(x)+C