What is the integral of (e^{tan(x)})/(1-sin^2(x)) ?

The integral of (e^{tan(x)})/(1-sin^2(x)) is e^{tan(x)}+C

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integral of (e^{tan(x)})/(1-sin^2(x))

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

e^{t a n (x)} + C

Solution steps

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

Rewrite using trig identities

= \int \frac{e ^{\frac{s i n ( x )}{c o s ( x )}}}{cos ^{2} ( x )} d x

Rewrite using trig identities

= \int e^{t a n (x)} sec^{2} (x) d x

Apply u-substitution

: \int e^{u} d u

= \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= e^{u}

Substitute back

u = tan (x)

= e^{t a n (x)}

Add a constant to the solution

= e^{t a n (x)} + C

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