What is the integral of x^2cos(1/3 x) ?

The integral of x^2cos(1/3 x) is 3x^2sin(1/3 x)-6(-3xcos(1/3 x)+9sin(1/3 x))+C

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integral of x^2cos(1/3 x)

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

3 x^{2} sin (\frac{1}{3} x) - 6 (- 3 x cos (\frac{1}{3} x) + 9 sin (\frac{1}{3} x)) + C

Solution steps

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

Apply Integration By Parts

: 3 x^{2} sin (\frac{1}{3} x) - \int 6 x sin (\frac{1}{3} x) d x

= 3 x^{2} sin (\frac{1}{3} x) - \int 6 x sin (\frac{1}{3} x) d x

\int 6 x sin (\frac{1}{3} x) d x = 6 (- 3 x cos (\frac{1}{3} x) + 9 sin (\frac{1}{3} x))

= 3 x^{2} sin (\frac{1}{3} x) - 6 (- 3 x cos (\frac{1}{3} x) + 9 sin (\frac{1}{3} x))

Add a constant to the solution

= 3 x^{2} sin (\frac{1}{3} x) - 6 (- 3 x cos (\frac{1}{3} x) + 9 sin (\frac{1}{3} x)) + C

\frac{d y}{d x} = - sin^{2} (x + y)

\int_{- 1}^{1} x + 2 d x

x \to 2 lim (\frac{2 - x}{2 x ^{2} - 4})

y = 9 x^{2}, y = x^{2} + 6

What is the integral of x^2cos(1/3 x) ?
The integral of x^2cos(1/3 x) is 3x^2sin(1/3 x)-6(-3xcos(1/3 x)+9sin(1/3 x))+C