What is the integral of (x^2-1)cos(2x) ?

The integral of (x^2-1)cos(2x) is 1/2 x^2sin(2x)+1/2 xcos(2x)-3/4 sin(2x)+C

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integral of (x^2-1)cos(2x)

\int (x^{2} - 1) cos (2 x) d x

\frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{3}{4} sin (2 x) + C

Solution steps

\int (x^{2} - 1) cos (2 x) d x

Expand

(x^{2} - 1) cos (2 x) : x^{2} cos (2 x) - cos (2 x)

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

\int x^{2} cos (2 x) d x = \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x)

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

= \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{1}{4} sin (2 x) - \frac{1}{2} sin (2 x)

Add similar elements:

- \frac{1}{4} sin (2 x) - \frac{1}{2} sin (2 x) = - \frac{3}{4} sin (2 x)

= \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{3}{4} sin (2 x)

Add a constant to the solution

= \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) - \frac{3}{4} sin (2 x) + C

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What is the integral of (x^2-1)cos(2x) ?
The integral of (x^2-1)cos(2x) is 1/2 x^2sin(2x)+1/2 xcos(2x)-3/4 sin(2x)+C