What is the integral of (sec^2(piw)+e^{-3w}) ?

The integral of (sec^2(piw)+e^{-3w}) is 1/pi tan(piw)-1/3 e^{-3w}+C

integral of (sec^2(piw)+e^{-3w})

\int (sec^{2} (π w) + e^{- 3 w}) d w

\frac{1}{π} tan (π w) - \frac{1}{3} e^{- 3 w} + C

Solution steps

\int sec^{2} (π w) + e^{- 3 w} d w

Apply the Sum Rule:

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

= \int sec^{2} (π w) d w + \int e^{- 3 w} d w

\int sec^{2} (π w) d w = \frac{1}{π} tan (π w)

\int e^{- 3 w} d w = - \frac{1}{3} e^{- 3 w}

= \frac{1}{π} tan (π w) - \frac{1}{3} e^{- 3 w}

Add a constant to the solution

= \frac{1}{π} tan (π w) - \frac{1}{3} e^{- 3 w} + C

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What is the integral of (sec^2(piw)+e^{-3w}) ?
The integral of (sec^2(piw)+e^{-3w}) is 1/pi tan(piw)-1/3 e^{-3w}+C