What is the integral from 0 to (2pi)/ω of tsin(ω)t ?

The integral from 0 to (2pi)/ω of tsin(ω)t is (8pi^3sin(ω))/(3ω^3)

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integral from 0 to (2pi)/ω of tsin(ω)t

\int_{0}^{\frac{2 π}{ω}} t sin (ω) t d t

\frac{8 π ^{3} sin ( ω )}{3 ω ^{3}}

Solution steps

\int_{0}^{\frac{2 π}{ω}} t sin (ω) t d t

Take the constant out:

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

= sin (ω) \cdot \int_{0}^{\frac{2 π}{ω}} tt d t

Simplify

tt : t^{2}

= sin (ω) \cdot \int_{0}^{\frac{2 π}{ω}} t^{2} d t

Apply the Power Rule

= sin (ω) [\frac{t ^{3}}{3}]_{0}^{\frac{2 π}{ω}}

Compute the boundaries:

\frac{8 π ^{3}}{3 ω ^{3}}

= sin (ω) \frac{8 π ^{3}}{3 ω ^{3}}

Simplify

= \frac{8 π ^{3} sin ( ω )}{3 ω ^{3}}

What is the integral from 0 to (2pi)/ω of tsin(ω)t ?
The integral from 0 to (2pi)/ω of tsin(ω)t is (8pi^3sin(ω))/(3ω^3)