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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int_{- \infty}^{0} 5^{x} d x

y = 9 - 5 x^{2}

\int_{2}^{4} \frac{x ^{2} + 8}{x ^{2}} d x

\int (\frac{6 x ^{5} 12 x ^{4} + 9 x ^{3}}{3 x ^{2}}) d x

\frac{1}{( s ^{2} + 1 ) ( s ^{2} + 4 s + 8 )}

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