What is the integral of pit-t^2cos(nt) ?

The integral of pit-t^2cos(nt) is (pit^2)/2-1/(n^3)(n^2t^2sin(nt)-2(-ntcos(nt)+sin(nt)))+C

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integral of pit-t^2cos(nt)

\int π t - t^{2} cos (n t) d t

\frac{π t ^{2}}{2} - \frac{1}{n ^{3}} (n^{2} t^{2} sin (n t) - 2 (- n t cos (n t) + sin (n t))) + C

Solution steps

\int π t - t^{2} cos (n t) d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int π t d t - \int t^{2} cos (n t) d t

\int π t d t = \frac{π t ^{2}}{2}

\int t^{2} cos (n t) d t = \frac{1}{n ^{3}} (n^{2} t^{2} sin (n t) - 2 (- n t cos (n t) + sin (n t)))

= \frac{π t ^{2}}{2} - \frac{1}{n ^{3}} (n^{2} t^{2} sin (n t) - 2 (- n t cos (n t) + sin (n t)))

Add a constant to the solution

= \frac{π t ^{2}}{2} - \frac{1}{n ^{3}} (n^{2} t^{2} sin (n t) - 2 (- n t cos (n t) + sin (n t))) + C

What is the integral of pit-t^2cos(nt) ?
The integral of pit-t^2cos(nt) is (pit^2)/2-1/(n^3)(n^2t^2sin(nt)-2(-ntcos(nt)+sin(nt)))+C