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

The integral of (x^2+3)cos(2x) is 1/2 x^2sin(2x)+1/2 xcos(2x)+5/4 sin(2x)+C

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

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

\frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) + \frac{5}{4} sin (2 x) + C

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

\frac{1}{2} sin (2 x) (x^{2} + 3) - (- \frac{1}{2} x cos (2 x) + \frac{1}{4} sin (2 x)) : \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) + \frac{5}{4} sin (2 x)

= \frac{1}{2} x^{2} sin (2 x) + \frac{1}{2} x cos (2 x) + \frac{5}{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{5}{4} sin (2 x) + C

What is the integral of (x^2+3)cos(2x) ?
The integral of (x^2+3)cos(2x) is 1/2 x^2sin(2x)+1/2 xcos(2x)+5/4 sin(2x)+C