# Integral of sec²(x)/exp(tan(x))

$\int&space;\frac{\sec^2(x)}{\exp(\tan(x))}\,\mathrm{d}x$

Although it may look a bit intimidating at first, I can guarantee you this is actually a very easy integral!

But what integration technique do you think will be useful? You might think we could perform integration by parts, but I really don’t think it’s a good idea to find the integral of sec²(x) or 1/exp(tan(x)). What we need is u-substitution. The “hard” part is to choose what to substitute. If you remember, the derivative of tan(x) is sec²(x), which we have at the numerator, and all we have to do is simply substitute sec²(x)dx with du and we will be left with 1/exp(u), which is way better!

$\int&space;\frac{\sec^2(x)}{\exp(\tan(x))}\,\mathrm{d}x$ ${\color{Blue}&space;u}={\color{Blue}&space;\tan(x)}\,\,\,\,\,{\color{Red}&space;\mathrm{d}u}={\color{Red}&space;\sec^2(x)\,\mathrm{d}x}$ $\int&space;\frac{{\color{Red}&space;\sec^2(x)}}{\exp({\color{Blue}\tan(x)})}{\color{Red}&space;\mathrm{d}x}$

Since sec²(x)dx is du we will write

$\int\frac{1}{\exp({\color{Blue}&space;u})}\,{\color{Red}&space;\mathrm{d}u}$

Let’s now talk about the function exp:

$\exp(x)=e^x$

and, in general,

$\exp(f(x))=e^{f(x)}$

This means

$\int&space;\frac{1}{\exp({\color{Blue}&space;u})}\,{\color{Red}&space;\mathrm{d}u}=\int\frac{1}{e^{\color{Blue}&space;u}}\,{\color{Red}&space;\mathrm{d}u}=\int&space;e^{-{\color{Blue}&space;u}}\,{\color{Red}&space;\mathrm{d}u}$

We know that the integral of e^u is equal to e^u, so this integral is –e^-u (simply apply the chain rule); therefore

$\int&space;e^{-{\color{Blue}&space;u}}\,{\color{Red}&space;\mathrm{d}u}=-e^{-{\color{Blue}&space;u}}$

At this point we can undo the substitution (remember that u=tan(x)):

$\int&space;e^{-{\color{Blue}&space;u}}\,{\color{Red}&space;\mathrm{d}u}=-e^{-{\color{Blue}&space;u}}\overset{{\color{Blue}&space;u}={\color{Blue}&space;\tan(x)}}{\rightarrow}=-e^{-{\color{Blue}&space;\tan(x)}}$

Overall:

$\int\frac{\sec^2(x)}{\exp(\tan(x))}\,\mathrm{d}x=-e^{-\tan(x)}+C$

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