What is the integral of e^x(1+tan(e^x)) ?

The integral of e^x(1+tan(e^x)) is e^x-ln|cos(e^x)|+C

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integral of e^x(1+tan(e^x))

\int e^{x} (1 + tan (e^{x})) d x

e^{x} - ln ∣ cos (e^{x}) ∣ + C

Solution steps

\int e^{x} (1 + tan (e^{x})) d x

Apply u-substitution

= \int 1 + tan (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u + \int tan (u) d u

\int 1 d u = u

\int tan (u) d u = - ln ∣ cos (u) ∣

= u - ln ∣ cos (u) ∣

Substitute back

u = e^{x}

= e^{x} - ln ∣ cos (e^{x}) ∣

Add a constant to the solution

= e^{x} - ln ∣ cos (e^{x}) ∣ + C

What is the integral of e^x(1+tan(e^x)) ?
The integral of e^x(1+tan(e^x)) is e^x-ln|cos(e^x)|+C