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

The integral of e^{-x}sec^2(e^{-x}) is -tan(e^{-x})+C

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

\int e^{- x} sec^{2} (e^{- x}) d x

- tan (e^{- x}) + C

Solution steps

\int e^{- x} sec^{2} (e^{- x}) d x

Apply u-substitution

: \int - sec^{2} (u) d u

= \int - sec^{2} (u) d u

Take the constant out:

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

= - \int sec^{2} (u) d u

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= - tan (u)

Substitute back

u = e^{- x}

= - tan (e^{- x})

Add a constant to the solution

= - tan (e^{- x}) + C

\frac{d}{d t} (sin (t) cos (t))

\frac{d y}{d x} = 2 y + 2

\int sin (3 x) sec^{2} (3 x) d x

\frac{d ^{2}}{d x ^{2}} (e^{- 4 x^{2}})

\frac{\partial}{\partial y} (\frac{x y ^{2}}{x + y})

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