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

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

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integral of (sec^2(pix)+e^{-3x})

\int (sec^{2} (π x) + e^{- 3 x}) d x

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

Solution steps

\int sec^{2} (π x) + e^{- 3 x} d x

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 x) d x + \int e^{- 3 x} d x

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

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

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

Add a constant to the solution

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

\int \frac{sin ( 2 π ) x}{3} d x

\int - \frac{e ^{x}}{x} d x

\int (e^{x}) \cdot sin (n x) d x

\int \frac{cos ( ln ( 4 x ) )}{x} d x

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