3. \n\\( \\int _ { 0 } ^ { \\frac { \\pi } { 4 } } \\frac { e ^ { \\tan x } } { \\cos ^ { 2 } x } d x \\)…

Explanation:

Step1: Use substitution

Let (u = \tan x), then (du=\frac{1}{\cos^{2}x}dx). When (x = 0), (u=\tan(0) = 0); when (x=\frac{\pi}{4}), (u=\tan(\frac{\pi}{4})=1). The integral (\int_{0}^{\frac{\pi}{4}}\frac{e^{\tan x}}{\cos^{2}x}dx) becomes (\int_{0}^{1}e^{u}du).

Step2: Integrate (e^{u})

The antiderivative of (e^{u}) is (e^{u}). By the fundamental theorem of calculus (\int_{0}^{1}e^{u}du=\left[e^{u}\right]_{0}^{1}).

Step3: Evaluate the definite - integral

[ \begin{align*} \left[e^{u}\right]_{0}^{1}&=e^{1}-e^{0}\ &=e - 1 \end{align*} ]

Answer:

C. (e - 1)