What is the integral of 3x^2sin(x/2) ?

The integral of 3x^2sin(x/2) is 3(-2x^2cos(x/2)+16(1/2 xsin(x/2)+cos(x/2)))+C

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integral of 3x^2sin(x/2)

\int 3 x^{2} sin (\frac{x}{2}) d x

3 (- 2 x^{2} cos (\frac{x}{2}) + 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))) + C

Solution steps

\int 3 x^{2} sin (\frac{x}{2}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x^{2} sin (\frac{x}{2}) d x

Apply Integration By Parts

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

= 3 (- 2 x^{2} cos (\frac{x}{2}) - \int - 4 x cos (\frac{x}{2}) d x)

\int - 4 x cos (\frac{x}{2}) d x = - 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))

= 3 (- 2 x^{2} cos (\frac{x}{2}) - (- 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))))

Simplify

= 3 (- 2 x^{2} cos (\frac{x}{2}) + 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2})))

Add a constant to the solution

= 3 (- 2 x^{2} cos (\frac{x}{2}) + 16 (\frac{1}{2} x sin (\frac{x}{2}) + cos (\frac{x}{2}))) + C

\int (x - e^{x}) d x

\int \frac{1}{x ^{2} + a ^{2}} d x

\frac{1}{s ^{2} - 2 s + 4}

x \frac{d y}{d x} - 4 y = 0

e^{4 x}

What is the integral of 3x^2sin(x/2) ?
The integral of 3x^2sin(x/2) is 3(-2x^2cos(x/2)+16(1/2 xsin(x/2)+cos(x/2)))+C