Question Number 17444 by Tinkutara last updated on 06/Jul/17
$$\mathrm{Evaluate}:\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\frac{{dx}}{\mathrm{cos}^{\mathrm{3}} \:{x}\:\sqrt{\mathrm{2}\:\mathrm{sin}\:\mathrm{2}{x}}} \\ $$
Answered by Arnab Maiti last updated on 10/Jul/17
![=∫_0 ^(π/4) (dx/(2(sinx)^(1/2) (cos x)^(7/2) )) =∫_0 ^(π/4) ((sec^4 x dx)/(2(tan x)^(1/2) )) =(1/2)∫_0 ^(π/4) (((1+tan^2 x)sec^2 x dx)/((tan x)^(1/2) )) put tanx=z ⇒sec^2 x dx=dz =(1/2)∫_0 ^( 1) (((1+z^2 ) dz)/z^(1/2) ) =(1/((√2) ))∫_0 ^( 1) (z^(−(1/2)) +z^(3/2) )dz =(1/2)[2z^(1/2) +(2/5)z^(5/2) ]_0 ^1 =(1/2)(2+(2/5))=1+(1/5)=(6/5)](Q17516.png)
$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{dx}}{\mathrm{2}\left(\mathrm{sinx}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{cos}\:\mathrm{x}\right)^{\frac{\mathrm{7}}{\mathrm{2}}} } \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\mathrm{sec}^{\mathrm{4}} \mathrm{x}\:\mathrm{dx}}{\mathrm{2}\left(\mathrm{tan}\:\mathrm{x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \mathrm{x}\right)\mathrm{sec}^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}}{\left(\mathrm{tan}\:\mathrm{x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} } \\ $$$$\:\:\mathrm{put}\:\mathrm{tanx}=\mathrm{z}\:\:\Rightarrow\mathrm{sec}^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}=\mathrm{dz} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left(\mathrm{1}+\mathrm{z}^{\mathrm{2}} \right)\:\mathrm{dz}}{\mathrm{z}^{\frac{\mathrm{1}}{\mathrm{2}}} } \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}\:}\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{z}^{−\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{z}^{\frac{\mathrm{3}}{\mathrm{2}}} \right)\mathrm{dz} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{2z}^{\frac{\mathrm{1}}{\mathrm{2}}} +\frac{\mathrm{2}}{\mathrm{5}}\mathrm{z}^{\frac{\mathrm{5}}{\mathrm{2}}} \right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2}+\frac{\mathrm{2}}{\mathrm{5}}\right)=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{5}}=\frac{\mathrm{6}}{\mathrm{5}} \\ $$
Commented by 786 last updated on 07/Jul/17
$$\mathrm{Your}\:\mathrm{answer}\:\mathrm{is}\:\mathrm{wrong}. \\ $$
Commented by Tinkutara last updated on 10/Jul/17
$$\mathrm{Yes}.\:\mathrm{Thanks}\:\mathrm{Sir}! \\ $$
Commented by Arnab Maiti last updated on 10/Jul/17
$$\mathrm{Is}\:\mathrm{it}\:\mathrm{right}\:\mathrm{now}\:? \\ $$