Question

If $\int\limits \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} = (\tan x)^A + C (\tan x )^B+ k ,$
where $k$ is a constant of integration, then $A + B + C$ equals :

• A. $\frac{21}{5}$
• B. $\frac{16}{5}$
• C. $\frac{07}{10}$
• D. $\frac{27}{10}$

Answer 1

Answer: (B)

$\int \frac{dx}{cos^{3}x\sqrt{4sin\, x\, cos\, x}}=\int \frac{dx}{2 cos^{4} x\sqrt{tan\, x}}$
Let tan $x=t^{2} \Rightarrow sec^{2}x=1+t^{4}$
$sec^{2}x\, dz =2t\, dt$
$\int \frac{sec^{4}x \,dx}{2\sqrt{tan x}}=\int \frac{sec^{2}x\left(sec^{2}x dx\right)}{2\sqrt{tan x}}=\int \frac{\left(1+t^{4}\right)2t dt}{2t}=\int\left(1+t^{4}\right)dt$
$=t+\frac{t^{5}}{5}+k =\sqrt{tan\, x }+\frac{1}{5}tan^{5/2}x+k\left[t=\sqrt{tan\, x}\right ]$
$A=\frac{1}{2},B=\frac{5}{2},C=\frac{1}{5} A+B+C=\frac{16}{5}$