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

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

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

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

Solution steps

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

Expand

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

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

Simplify

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

\int r^{2} (\frac{r ^{3}}{18} + 1)^{5} d r

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\int (2 x^{5} - 20 x)^{3} (x^{4} - 2) d x

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

\int - 11 sin (11 x) d x

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