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

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

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

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

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

Solution steps

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

Simplify

\frac{1}{2} x : \frac{x}{2}

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

Apply Integration By Parts

: 2 x^{2} sin (\frac{x}{2}) - \int 4 x sin (\frac{x}{2}) d x

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

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

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

Add a constant to the solution

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

\int_{0}^{1} \frac{6}{1 + t ^{2}} d t

\frac{d}{d x} (\frac{cos ( x )}{x} + \frac{x}{cos ( x )})

\frac{1}{4 x ^{2} + 1}

x \to 3 - lim (e^{\frac{3}{3 - x}})

x \to 0 + lim (\frac{x ^{2}}{9} - \frac{2}{x})

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