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

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

Popular >

integral of (e^{-x})/(cos^2(e^{-x))}

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

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

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \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

ln (x^{2} - 7)

\int \frac{x - 1}{1 - x ^{2}} d x

\frac{d}{d x} f (x) = x^{2} + 8 x + 7

x \to 0 lim (\frac{7 x ^{5} + 5 x ^{2} + 12 x}{3 x ^{5} + 4 x})

x^{2} - 3 x y = 10

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