What is the integral from 0 to 2pi of 6x^2sin(x/4) ?

The integral from 0 to 2pi of 6x^2sin(x/4) is 384(pi-2)

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integral from 0 to 2pi of 6x^2sin(x/4)

\int_{0}^{2 π} 6 x^{2} sin (\frac{x}{4}) d x

384 (π - 2)

Decimal

438.37157 \dots

Solution steps

\int_{0}^{2 π} 6 x^{2} sin (\frac{x}{4}) d x

Take the constant out:

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

= 6 \cdot \int_{0}^{2 π} x^{2} sin (\frac{x}{4}) d x

Apply u-substitution

= 6 \cdot \int_{0}^{\frac{π}{2}} 64 u^{2} sin (u) d u

Take the constant out:

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

= 6 \cdot 64 \cdot \int_{0}^{\frac{π}{2}} u^{2} sin (u) d u

Apply Integration By Parts

= 6 \cdot 64 [- u^{2} cos (u) - \int - 2 u cos (u) d u]_{0}^{\frac{π}{2}}

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

= 6 \cdot 64 [- u^{2} cos (u) - (- 2 (u sin (u) + cos (u)))]_{0}^{\frac{π}{2}}

Simplify

= 384 [- u^{2} cos (u) + 2 (u sin (u) + cos (u))]_{0}^{\frac{π}{2}}

Compute the boundaries:

π - 2

= 384 (π - 2)

What is the integral from 0 to 2pi of 6x^2sin(x/4) ?
The integral from 0 to 2pi of 6x^2sin(x/4) is 384(pi-2)