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

The integral of 2x^2sin(x/3) is 54(-1/9 x^2cos(x/3)+2(1/3 xsin(x/3)+cos(x/3)))+C

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int 27 u^{2} sin (u) d u

= 2 \cdot \int 27 u^{2} sin (u) d u

Take the constant out:

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

= 2 \cdot 27 \cdot \int u^{2} sin (u) d u

Apply Integration By Parts

: - u^{2} cos (u) - \int - 2 u cos (u) d u

= 2 \cdot 27 (- u^{2} cos (u) - \int - 2 u cos (u) d u)

\int - 2 u cos (u) d u = - 2 (u sin (u) + cos (u))

= 2 \cdot 27 (- u^{2} cos (u) - (- 2 (u sin (u) + cos (u))))

Substitute back

u = \frac{x}{3}

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

Simplify

2 \cdot 27 (- (\frac{x}{3})^{2} cos (\frac{x}{3}) - (- 2 (\frac{x}{3} sin (\frac{x}{3}) + cos (\frac{x}{3})))) : 54 (- \frac{1}{9} x^{2} cos (\frac{x}{3}) + 2 (\frac{1}{3} x sin (\frac{x}{3}) + cos (\frac{x}{3})))

= 54 (- \frac{1}{9} x^{2} cos (\frac{x}{3}) + 2 (\frac{1}{3} x sin (\frac{x}{3}) + cos (\frac{x}{3})))

Add a constant to the solution

= 54 (- \frac{1}{9} x^{2} cos (\frac{x}{3}) + 2 (\frac{1}{3} x sin (\frac{x}{3}) + cos (\frac{x}{3}))) + C

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What is the integral of 2x^2sin(x/3) ?
The integral of 2x^2sin(x/3) is 54(-1/9 x^2cos(x/3)+2(1/3 xsin(x/3)+cos(x/3)))+C