If $f\left(x\right)=\frac{\left(2^x+2^{-x}\right)\left(tanx\right)\sqrt[]{ta{n}^{-1}\left(2x^2-3x+1\right)}}{{\left(7x^2-3x+1\right)}^3}$ , then f′(0) is equal to

It's difficult but in some colleges you may can get 

  $\overrightarrow{a}+5\overrightarrow{b}$  is collinear with $\overrightarrow{c}$  

⇒   $\overrightarrow{a}+5\overrightarrow{b}$ = $\overrightarrow{c}$           …(1)

$\overrightarrow{b}+6\overrightarrow{c}$ is collinear with $\overrightarrow{a}$  

⇒   $\overrightarrow{b}+6\overrightarrow{c}=\mu \overrightarrow{a}$               …(2)

From (1) and (2)

  $\overrightarrow{b}+6\overrightarrow{c}=\mu \left(\lambda \overrightarrow{c}-5\overrightarrow{b}\right)$          

-> $\left(1+5\mu \right)\overrightarrow{b}+\left(6-\lambda \mu \right)\overrightarrow{c}=0$

$?\overrightarrow{b}$ and $\overrightarrow{c}$

${\displaystyle \int \frac{\left(sinx-cosx\right)si{n}^{2}x}{tanx\left(si{n}^{3}x+co{s}^{3}x\right)}dx}$

${\displaystyle \int \frac{\left(sinx-cosx\right)sinxcosx}{si{n}^{3}x+co{s}^{3}x}dx}$ , put sin³x + cos³x = t(3 sin²x×cosx – 3cos²xsinx) dx = dt

-> $\frac{1}{3}{\displaystyle \int \frac{dt}{t}}$

$=\frac{lnt}{3}+c$

$=\frac{ln\left|si{n}^{3}x+co{s}^{3}x\right|}{3}+c$

$Area=\frac{\pi {\left(13\right)}^2}{2}-\left[\frac{1}{2}\times 25\times 5+{\displaystyle \underset{12}{\overset{13}{\int }}\sqrt[]{\left(169-y^2\right)}}\cdot dy\right]$

$=\frac{169\pi }{2}-\left[\frac{125}{2}+{\left[\frac{y}{2}\sqrt[]{169-y^2}+\frac{169}{2}si{n}^{-1}\frac{y}{13}\right]}_{12}^{13}\right]$

$=\frac{169}{2}\pi -\frac{125}{2}-\left[\frac{169}{2}\times \frac{\pi }{2}-6\times 5-\frac{169}{2}si{n}^{-1}\frac{12}{13}\right]$

$=\frac{169\pi }{4}-\frac{65}{2}+\frac{169}{2}si{n}^{-1}\frac{12}{13}$