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

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

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

Apply Integration By Parts

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

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

Simplify

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

\frac{\partial}{\partial x} (x^{(x^{2} + y^{2})})

\int (5 - 3 x)^{10} d x

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

e^{\frac{x}{y}} = 5 x - y

(t - 2)^{2}

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