Question

$\huge{ \boxed{ \bold \red{ \int \tan(x) dx}}}$


❖ᴏɴʟʏ ᴘʀᴏᴘᴇʀ ꜱᴏʟᴠᴇᴅ ᴀɴꜱᴡᴇʀ ᴡɪᴛʜ ɢᴏᴏᴅ ᴇxᴘʟᴀɴᴀɪᴏɴ ɴᴇᴇᴅᴇᴅ
❖ ɴᴏ ꜱᴘᴀᴍᴍɪɴɢ
❖ᴏɴʟʏ ꜰᴏʀ ᴍᴏᴅᴇʀᴀᴛᴏʀꜱ, ʙʀᴀɪɴʟʏ ꜱᴛᴀʀꜱ ᴀɴᴅ ᴏᴛʜᴇʀ ʙᴇꜱᴛ ᴜꜱᴇʀꜱ​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{\displaystyle\int\,tan(x)\,dx}$

$\sf{=\displaystyle\int\,\dfrac{sin(x)}{cos(x)}\,dx}$

$\sf{Let\,\,\,cos(x)=t}$

$\sf{\implies\,-sin(x)\,dx=dt}$

$\sf{=-\displaystyle\int\,\dfrac{1}{t}\,dt}$

$\sf{=-\ln|t|+C}$

$\sf{=-\ln|cos(x)|+C}$

$\sf{=\ln|cos(x)|^{-1}+C}$

$\sf{=\ln\left|\dfrac{1}{cos(x)}\right|+C}$

$\sf{=\ln\left|sec(x)\right|+C}$

Answer 2

$\bold \red{\int{\tan{\left(x \right)} d x}}$

$\bold \red{Rewrite \: the \: tangent \: as }$

$\bold \red{\tan\left(x\right)=\frac{\sin\left(x\right)}{\cos\left(x\right)}:}$

$\bold{\color{red}{\int{\tan{\left(x \right)} d x}} = \color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}}}$

$\bold \red{Let \: u=\cos{\left(x \right)}.}$

${ \bold \red{Then \: du=\left(\cos{\left(x \right)}\right)^{\prime }dx = - \sin{\left(x \right)} dx \: and \: we \: have \: that \sin{\left(x \right)} dx = - du.}}$

$\bold \red{Thus,}$

$\bold \red{\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}} = \color{red}{\int{\left(- \frac{1}{u}\right)d u}}}$

$\bold \red{Apply \: the \: constant \: multiple \: rule}$

$\bold \red{{\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du \: with \: c=-1 \: and \: f{\left(u \right)} = \frac{1}{u}:}}$

$\bold{\color{red}{\int{\left(- \frac{1}{u}\right)d u}} = \color{red}{\left(- \int{\frac{1}{u} d u}\right)} \: The \: integral \: of \: \frac{1}{u} \: is \: \int{\frac{1}{u} d u} = \ln{\left(u \right)}}$

$\bold \red{- \color{red}{\int{\frac{1}{u} d u}} = - \color{red}{\ln{\left(u \right)}}}$

$\bold \red{Recall \: that \: u=\cos{\left(x \right)}:}$

$\bold \red{- \ln{\left(\color{red}{u} \right)} = - \ln{\left(\color{red}{\cos{\left(x \right)}} \right)}}$

$\bold \red{Therefore,}$

$\bold \red{\int{\tan{\left(x \right)} d x} = - \ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}}$

$\bold \red{Add \: the \: constant \: of \: integration:}$

${\bold \red{\int{\tan{\left(x \right)} d x} = - \ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}+C}}$