What is the integral of (tan(x)e^{sec(x)})/(cos(x)) ?

The integral of (tan(x)e^{sec(x)})/(cos(x)) is e^{sec(x)}+C

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integral of (tan(x)e^{sec(x)})/(cos(x))

\int \frac{tan ( x ) e ^{s e c (x)}}{cos ( x )} d x

e^{s e c (x)} + C

Solution steps

\int \frac{tan ( x ) e ^{s e c (x)}}{cos ( x )} d x

Rewrite using trig identities

= \int \frac{e ^{s e c (x)} sin ( x )}{cos ^{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 = sec (x)

= e^{s e c (x)}

Add a constant to the solution

= e^{s e c (x)} + C

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