Question

∫ eˣ sec x(1+tanx) dx=........+c,Select correct option from the given options.
(a) eˣ sec x tan x
(b) eˣ tan x
(c) eˣ sec x
(d) -eˣ sec x

Answer 1

HELLO DEAR,



GIVEN:-
∫e^xsecx(1 + tanx).dx

∫e^x(secx + secxtanx).dx

we know if the integral is in the form of e^x{f(x) + f'(x)} then your integral is e^xf(x) + c.


therefore, ∫e^x(secx + secxtanx).dx = e^xsecx + c.


I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

we have to find the value of $\int{e^xsecx(1+tanx)}\,dx$

$\int{e^xsecx(1+tanx)}\,dx$

= $\int{e^x\{secx.1+secx.tanx\}}\,dx$

= $\int{e^x\{\underbrace{secx}+\underbrace{sec.tanx}\}}\,dx$

if we assume secx = f(x)
then, secx.tanx = f'(x)
and we know that $\int{e^x\{f(x)+f'(x)\}}\,dx$ =$e^xf(x)+C$

so, $\int{e^x\{secx+secx.tanx\}}\,dx=e^xsecx+C$

hence, option (c) is correct.