What is the integral of 5tcos(pi*t) ?

The integral of 5tcos(pi*t) is 5(1/pi tsin(pit)+1/(pi^2)cos(pit))+C

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integral of 5tcos(pi*t)

\int 5 t cos (π \cdot t) d t

5 (\frac{1}{π} t sin (π t) + \frac{1}{π ^{2}} cos (π t)) + C

Solution steps

\int 5 t cos (π t) d t

Take the constant out:

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

= 5 \cdot \int t cos (π t) d t

Apply Integration By Parts

: \frac{1}{π} t sin (π t) - \int \frac{1}{π} sin (π t) d t

= 5 (\frac{1}{π} t sin (π t) - \int \frac{1}{π} sin (π t) d t)

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

= 5 (\frac{1}{π} t sin (π t) - (- \frac{1}{π ^{2}} cos (π t)))

Simplify

= 5 (\frac{1}{π} t sin (π t) + \frac{1}{π ^{2}} cos (π t))

Add a constant to the solution

= 5 (\frac{1}{π} t sin (π t) + \frac{1}{π ^{2}} cos (π t)) + C

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What is the integral of 5tcos(pi*t) ?
The integral of 5tcos(pi*t) is 5(1/pi tsin(pit)+1/(pi^2)cos(pit))+C