Question

$\int \: \frac{ { \sec}^{2}(x )}{ \tan(x) } dx$
integration .....​

Answer 1

$\huge \boxed{ \red{\mathfrak{solution}}}$

$\int \: \frac{ \sec^{2}x }{tanx} dx$

Here we use substitution method :

Let tanx = t

=> sec^2x dx = dt

put

sec^2x= dt

and

tanx= t

in Given question

$\rightarrow\int \frac{dt}{t} \\ \\ \rightarrow \: log(t)$

Now put the value of t = tanx

$\rightarrow \: log(tanx) + c$

Formula used :

$\boxed{ \green{ \bold{\frac{d}{dx} tanx = {sec}^{2} x}}}$

and

$\boxed{ \purple{ \bold{\int \: \frac{1}{x} = log(x) }}}$

Answer 2

$\huge{\fbox{\fbox{\bigstar{\mathfrak{\red{Answer}}}}}}$

Here we use substitution method :

Let tanx = t

=> sec^2x dx = dt

put

sec^2x= dt

and

tanx= t

in Given question

\rightarrow\int \frac{dt}{t} \\ \\ \rightarrow \: log(t)

Now put the value of t = tanx

\rightarrow \: log(tanx) + c

Formula used :

\boxed{ \green{ \bold{\frac{d}{dx} tanx = {sec}^{2} x}}}

and

\boxed{ \purple{ \bold{\int \: \frac{1}{x} = log(x) }}}