Question

If $\int \frac{dx}{{\cos }^{3}x\sqrt{2\sin 2x}}=(\tan x{)}^A+C(\tan x{)}^B+k,$
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)

$\int \frac{dx}{co{s}^{3}x\sqrt{4sinxcosx}}=\int \frac{dx}{2co{s}^{4}x\sqrt{tanx}}$
Let tan $x=t^2\Rightarrow se{c}^{2}x=1+t^4$
$se{c}^{2}xdz=2tdt$
$\int \frac{se{c}^{4}xdx}{2\sqrt{tanx}}=\int \frac{se{c}^{2}x\left(se{c}^{2}xdx\right)}{2\sqrt{tanx}}=\int \frac{\left(1+t^4\right)2tdt}{2t}=\int \left(1+t^4\right)dt$
$=t+\frac{t^5}{5}+k=\sqrt{tanx}+\frac{1}{5}ta{n}^{5/2}x+k\left[t=\sqrt{tanx}\right]$
$A=\frac{1}{2},B=\frac{5}{2},C=\frac{1}{5}A+B+C=\frac{16}{5}$