What is the integral of (2x^2-4x-3)cos(2x) ?

The integral of (2x^2-4x-3)cos(2x) is x^2sin(2x)+xcos(2x)-2xsin(2x)-2sin(2x)-cos(2x)+C

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integral of (2x^2-4x-3)cos(2x)

\int (2 x^{2} - 4 x - 3) cos (2 x) d x

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

Solution steps

\int (2 x^{2} - 4 x - 3) cos (2 x) d x

Expand

(2 x^{2} - 4 x - 3) cos (2 x) : 2 x^{2} cos (2 x) - 4 x cos (2 x) - 3 cos (2 x)

= \int 2 x^{2} cos (2 x) - 4 x cos (2 x) - 3 cos (2 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 2 x^{2} cos (2 x) d x - \int 4 x cos (2 x) d x - \int 3 cos (2 x) d x

\int 2 x^{2} cos (2 x) d x = x^{2} sin (2 x) + x cos (2 x) - \frac{1}{2} sin (2 x)

\int 4 x cos (2 x) d x = 2 x sin (2 x) + cos (2 x)

\int 3 cos (2 x) d x = \frac{3}{2} sin (2 x)

= x^{2} sin (2 x) + x cos (2 x) - \frac{1}{2} sin (2 x) - (2 x sin (2 x) + cos (2 x)) - \frac{3}{2} sin (2 x)

Simplify

x^{2} sin (2 x) + x cos (2 x) - \frac{1}{2} sin (2 x) - (2 x sin (2 x) + cos (2 x)) - \frac{3}{2} sin (2 x) : x^{2} sin (2 x) + x cos (2 x) - 2 x sin (2 x) - 2 sin (2 x) - cos (2 x)

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

Add a constant to the solution

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

\int x x^{2} + 4 d x

\int 400 - 16 x^{2} d x

\int \frac{32}{x + x x} d x

\int \frac{x}{5 x} d x

\int x^{3} + e^{x} d x

What is the integral of (2x^2-4x-3)cos(2x) ?
The integral of (2x^2-4x-3)cos(2x) is x^2sin(2x)+xcos(2x)-2xsin(2x)-2sin(2x)-cos(2x)+C