What is the integral from 0 to 2pi of tsin(nt) ?

The integral from 0 to 2pi of tsin(nt) is (sin(2pin)-2pincos(2pin))/(n^2)

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integral from 0 to 2pi of tsin(nt)

\int_{0}^{2 π} t sin (n t) d t

\frac{sin ( 2 πn ) - 2 πn cos ( 2 πn )}{n ^{2}}

Solution steps

\int_{0}^{2 π} t sin (n t) d t

Apply Integration By Parts

= [\frac{1}{n} (- t cos (n t) - n \cdot \int - \frac{1}{n} cos (n t) d t)]_{0}^{2 π}

\int - \frac{1}{n} cos (n t) d t = - \frac{1}{n ^{2}} sin (n t)

= [\frac{1}{n} (- t cos (n t) - n (- \frac{1}{n ^{2}} sin (n t)))]_{0}^{2 π}

Simplify

[\frac{1}{n} (- t cos (n t) - n (- \frac{1}{n ^{2}} sin (n t)))]_{0}^{2 π} : [\frac{1}{n} (- t cos (n t) + \frac{1}{n} sin (n t))]_{0}^{2 π}

= [\frac{1}{n} (- t cos (n t) + \frac{1}{n} sin (n t))]_{0}^{2 π}

Compute the boundaries:

\frac{1}{n} (\frac{1}{n} sin (2 πn) - 2 π cos (2 πn))

= \frac{1}{n} (\frac{1}{n} sin (2 πn) - 2 π cos (2 πn))

Simplify

= \frac{sin ( 2 πn ) - 2 πn cos ( 2 πn )}{n ^{2}}

What is the integral from 0 to 2pi of tsin(nt) ?
The integral from 0 to 2pi of tsin(nt) is (sin(2pin)-2pincos(2pin))/(n^2)