What is the integral of e^xsec(e^x) ?

The integral of e^xsec(e^x) is ln|tan(e^x)+sec(e^x)|+C

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integral of e^xsec(e^x)

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

ln ∣ tan (e^{x}) + sec (e^{x}) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

: \int sec (u) d u

= \int sec (u) d u

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= ln ∣ tan (u) + sec (u) ∣

Substitute back

u = e^{x}

= ln ∣ tan (e^{x}) + sec (e^{x}) ∣

Add a constant to the solution

= ln ∣ tan (e^{x}) + sec (e^{x}) ∣ + C

\int tan^{5} (7 x) d x

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x \to 2 x lim (sin (x))

What is the integral of e^xsec(e^x) ?
The integral of e^xsec(e^x) is ln|tan(e^x)+sec(e^x)|+C