What is the integral of (e^{tan(θ)})/(cos^2(θ)) ?

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

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integral of (e^{tan(θ)})/(cos^2(θ))

\int \frac{e ^{t a n (θ)}}{cos ^{2} ( θ )} d θ

e^{t a n (θ)} + C

Solution steps

\int \frac{e ^{t a n (θ)}}{cos ^{2} ( θ )} d θ

Rewrite using trig identities

= \int \frac{e ^{t a n (θ)}}{1 - sin ^{2} ( θ )} d θ

Apply Integration By Parts

= e^{t a n (θ)} tan (θ) - \int e^{t a n (θ)} sec^{2} (θ) tan (θ) d θ

\int e^{t a n (θ)} sec^{2} (θ) tan (θ) d θ = tan (θ) e^{t a n (θ)} - e^{t a n (θ)}

= e^{t a n (θ)} tan (θ) - (tan (θ) e^{t a n (θ)} - e^{t a n (θ)})

Simplify

e^{t a n (θ)} tan (θ) - (tan (θ) e^{t a n (θ)} - e^{t a n (θ)}) : e^{t a n (θ)}

= e^{t a n (θ)}

Add a constant to the solution

= e^{t a n (θ)} + C

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