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

The integral of 3x^2sin(3x) is -x^2cos(3x)+2/3 xsin(3x)+2/9 cos(3x)+C

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integral of 3x^2sin(3x)

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

- x^{2} cos (3 x) + \frac{2}{3} x sin (3 x) + \frac{2}{9} cos (3 x) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{2}{3} x cos (3 x) d x = - \frac{2}{27} (3 x sin (3 x) + cos (3 x))

= 3 (- \frac{1}{3} x^{2} cos (3 x) - (- \frac{2}{27} (3 x sin (3 x) + cos (3 x))))

Simplify

3 (- \frac{1}{3} x^{2} cos (3 x) - (- \frac{2}{27} (3 x sin (3 x) + cos (3 x)))) : - x^{2} cos (3 x) + \frac{2}{3} x sin (3 x) + \frac{2}{9} cos (3 x)

= - x^{2} cos (3 x) + \frac{2}{3} x sin (3 x) + \frac{2}{9} cos (3 x)

Add a constant to the solution

= - x^{2} cos (3 x) + \frac{2}{3} x sin (3 x) + \frac{2}{9} cos (3 x) + C

f (x) = 3^{x^{2} - 3}

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

y = (3 x - x^{2}) (3 - x - x^{2})

3 y

\frac{d}{d x} (- 3 x y ln (x y))

What is the integral of 3x^2sin(3x) ?
The integral of 3x^2sin(3x) is -x^2cos(3x)+2/3 xsin(3x)+2/9 cos(3x)+C