What is the integral of 216x^2cos(6x) ?

The integral of 216x^2cos(6x) is 36x^2sin(6x)-2(-6xcos(6x)+sin(6x))+C

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integral of 216x^2cos(6x)

\int 216 x^{2} cos (6 x) d x

36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x)) + C

Solution steps

\int 216 x^{2} cos (6 x) d x

Take the constant out:

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

= 216 \cdot \int x^{2} cos (6 x) d x

Apply u-substitution

: \int \frac{u ^{2} cos ( u )}{216} d u

= 216 \cdot \int \frac{u ^{2} cos ( u )}{216} d u

Take the constant out:

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

= 216 \cdot \frac{1}{216} \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

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

= 216 \cdot \frac{1}{216} (u^{2} sin (u) - \int 2 u sin (u) d u)

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

= 216 \cdot \frac{1}{216} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Substitute back

u = 6 x

= 216 \cdot \frac{1}{216} ((6 x)^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x)))

Simplify

216 \cdot \frac{1}{216} ((6 x)^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x))) : 36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x))

= 36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x))

Add a constant to the solution

= 36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x)) + C

\frac{\partial}{\partial y} (x^{\frac{2}{3}} y^{\frac{1}{3}})

x \to 3 lim (- \frac{3}{7} x + \frac{4}{11})

x^{2} e^{x} - 2 x e^{x} + 2 e^{x}

y^{^{''}} - 10 y^{^{'}} + 26 y = 4

\int \frac{4}{4 x ^{2} + 4 x + 65} d x

What is the integral of 216x^2cos(6x) ?
The integral of 216x^2cos(6x) is 36x^2sin(6x)-2(-6xcos(6x)+sin(6x))+C