What is the integral of pitcos(nt) ?

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

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integral of pitcos(nt)

\int π t cos (n t) d t

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

Solution steps

\int π t cos (n t) d t

Take the constant out:

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

= π \cdot \int t cos (n t) d t

Apply u-substitution

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

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

Take the constant out:

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

= π \frac{1}{n ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

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

= π \frac{1}{n ^{2}} (u sin (u) - \int sin (u) d u)

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

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

Substitute back

u = n t

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

Simplify

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

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

Add a constant to the solution

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

x \to 0 lim (\frac{2 x}{e ^{x} - 1})

\int \frac{1}{x ^{3} - 216} d x

x = - 2, x = 1, y = 10 x, y = x^{2} - 11

\frac{d}{d x} (\frac{x ^{4} - x ^{3} - 8}{( x - 3 ) ( x + 2 )})

y = x^{3} - 4, (2, 0)

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