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

The integral of e^{2x}sec(e^{2x}) is 1/2 ln|tan(e^{2x})+sec(e^{2x})|+C

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

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

\frac{1}{2} ln tan (e^{2 x}) + sec (e^{2 x}) + C

Solution steps

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

Apply u-substitution

: \int \frac{sec ( u )}{2} d u

= \int \frac{sec ( u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = e^{2 x}

= \frac{1}{2} ln tan (e^{2 x}) + sec (e^{2 x})

Add a constant to the solution

= \frac{1}{2} ln tan (e^{2 x}) + sec (e^{2 x}) + C

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What is the integral of e^{2x}sec(e^{2x}) ?
The integral of e^{2x}sec(e^{2x}) is 1/2 ln|tan(e^{2x})+sec(e^{2x})|+C