Question

Evaluate the integral of (sec a tan a/(√e^(seca))) da from 0 to π/3

Answer 1

$I$ $=\int_{0}^{\pi/3} \dfrac{seca\times tana}{\sqrt{e^{seca}}}da$

$\rightarrow \int_{0}^{\pi/3} (e^{\dfrac{-seca}{2}})(seca\times tana )da$

Substitute, u= $e^{\dfrac{-seca}{2}}$

$Then, \dfrac{du}{da}=- \dfrac{seca\times tana}{2} \rightarrow da=- \dfrac{2}{seca\times tana}du$

On substituting, we get

$\int\dfrac{seca\times tana}{\sqrt{e^{seca}}}da=-2 \int e^udu$ + C

$= -2 e^u +C$

Now put the value of u=$e^{\dfrac{-seca}{2}}$ in above equation,

$I= \mid-2e^{-\dfrac{seca}{2}}\mid_{0}^{\pi/3}$

On solving limit, we get

$I=\dfrac{2}{\sqrt e} - 2e^{-1}$

On simplification,

$I=e^{-1}(2\sqrt e-2)$

Hence the value of $\int_{0}^{\pi/3} \dfrac{seca\times tana}{\sqrt{e^{seca}}}da=$ $e^{-1}(2\sqrt e-2)$

Approximation:

$I=0.4773024370823822$