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

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

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integral of (tan(x))/(cos^2(x))e^{tan(x)}

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

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

Solution steps

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

Simplify

\frac{tan ( x )}{cos ^{2} ( x )} e^{t a n (x)} : e^{t a n (x)} sec^{2} (x) tan (x)

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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