What is the integral of 512x^2cos(8x) ?

The integral of 512x^2cos(8x) is 64x^2sin(8x)-2(-8xcos(8x)+sin(8x))+C

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integral of 512x^2cos(8x)

\int 512 x^{2} cos (8 x) d x

64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x)) + C

Solution steps

\int 512 x^{2} cos (8 x) d x

Take the constant out:

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

= 512 \cdot \int x^{2} cos (8 x) d x

Apply u-substitution

: \int \frac{u ^{2} cos ( u )}{512} d u

= 512 \cdot \int \frac{u ^{2} cos ( u )}{512} d u

Take the constant out:

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

= 512 \cdot \frac{1}{512} \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

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

= 512 \cdot \frac{1}{512} (u^{2} sin (u) - \int 2 u sin (u) d u)

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

= 512 \cdot \frac{1}{512} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Substitute back

u = 8 x

= 512 \cdot \frac{1}{512} ((8 x)^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x)))

Simplify

512 \cdot \frac{1}{512} ((8 x)^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x))) : 64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x))

= 64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x))

Add a constant to the solution

= 64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x)) + C

\frac{d}{d x} (ln (\frac{sin ( x )}{x}))

y = 4 ln (x), y = x ln (x)

x \to 4 lim (e^{\frac{3}{4 - x}})

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

\int 3 sin (ln (x)) d x

What is the integral of 512x^2cos(8x) ?
The integral of 512x^2cos(8x) is 64x^2sin(8x)-2(-8xcos(8x)+sin(8x))+C