What is the integral of tcos(n pi/2 t) ?

The integral of tcos(n pi/2 t) is 4/(pi^2n^2)(pi/2 ntsin(pi/2 nt)+cos(pi/2 nt))+C

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integral of tcos(n pi/2 t)

\int t cos (n \frac{π}{2} t) d t

\frac{4}{π ^{2} n ^{2}} (\frac{π}{2} n t sin (\frac{π}{2} n t) + cos (\frac{π}{2} n t)) + C

Solution steps

\int t cos (n \frac{π}{2} t) d t

\frac{π}{2} = π \frac{1}{2}

= \int t cos (nπ \frac{1}{2} t) d t

Apply u-substitution

: \int \frac{4 u cos ( u )}{π ^{2} n ^{2}} d u

= \int \frac{4 u cos ( u )}{π ^{2} n ^{2}} d u

Take the constant out:

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

= \frac{4}{π ^{2} n ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{4}{π ^{2} n ^{2}} (u sin (u) - \int sin (u) d u)

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

= \frac{4}{π ^{2} n ^{2}} (u sin (u) - (- cos (u)))

Substitute back

u = nπ \frac{1}{2} t

= \frac{4}{π ^{2} n ^{2}} (nπ \frac{1}{2} t sin (nπ \frac{1}{2} t) - (- cos (nπ \frac{1}{2} t)))

Simplify

\frac{4}{π ^{2} n ^{2}} (nπ \frac{1}{2} t sin (nπ \frac{1}{2} t) - (- cos (nπ \frac{1}{2} t))) : \frac{4}{π ^{2} n ^{2}} (\frac{π}{2} n t sin (\frac{π}{2} n t) + cos (\frac{π}{2} n t))

= \frac{4}{π ^{2} n ^{2}} (\frac{π}{2} n t sin (\frac{π}{2} n t) + cos (\frac{π}{2} n t))

Add a constant to the solution

= \frac{4}{π ^{2} n ^{2}} (\frac{π}{2} n t sin (\frac{π}{2} n t) + cos (\frac{π}{2} n t)) + C

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What is the integral of tcos(n pi/2 t) ?
The integral of tcos(n pi/2 t) is 4/(pi^2n^2)(pi/2 ntsin(pi/2 nt)+cos(pi/2 nt))+C