Question Number 140548 by liberty last updated on 09/May/21
$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\begin{vmatrix}{\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\:\:\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\:\:\:\:\:\mathrm{csc}^{\mathrm{2}} \mathrm{x}}\\{\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\:\:\:\:\:\mathrm{tan}\:\:^{\mathrm{2}} \mathrm{x}}\end{vmatrix} \\ $$$$\mathrm{evaluate}\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}. \\ $$
Answered by EDWIN88 last updated on 09/May/21
![I=∫_0 ^(π/4) f(x) dx = determinant (((∫_0 ^(π/4) sec^2 x dx 1 1)),((∫_0 ^(π/4) cos^2 x dx ∫_0 ^(π/4) cos^2 x dx ∫_0 ^(π/4) csc^2 x dx)),(( 1 ∫_0 ^(π/4) cos^2 x dx ∫_0 ^(π/4) tan^2 x dx))) (1)∫_0 ^(π/4) sec^2 x dx= 1 (2) ∫_0 ^(π/4) cos^2 x dx = ∫_0 ^(π/4) (((1+cos 2x)/2))dx=[((x+((sin 2x)/2))/2) ]_0 ^(π/4) =((π+2)/8) (3)∫_0 ^(π/4) csc^2 x dx=−1 (4)∫_0 ^(π/4) tan^2 x dx=∫_0 ^(π/4) (sec^2 x−1)dx=[ tan x−x ]_0 ^(π/4) =((4−π)/4) I= determinant ((( 1 1 1 )),((((π+2)/8) ((π+2)/8) −1)),(( 1 ((π+2)/8) ((4−π)/4))))](https://www.tinkutara.com/question/Q140549.png)
$$\mathrm{I}=\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\begin{vmatrix}{\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\:\:\:\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\:\:\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{csc}^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}}\\{\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\:\:\:\:\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{tan}^{\mathrm{2}} \:\mathrm{x}\:\mathrm{dx}}\end{vmatrix} \\ $$$$\left(\mathrm{1}\right)\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}=\:\mathrm{1}\:\:\:\:\left(\mathrm{2}\right)\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}\:=\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\left(\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{2x}}{\mathrm{2}}\right)\mathrm{dx}=\left[\frac{\mathrm{x}+\frac{\mathrm{sin}\:\mathrm{2x}}{\mathrm{2}}}{\mathrm{2}}\:\right]_{\mathrm{0}} ^{\pi/\mathrm{4}} =\frac{\pi+\mathrm{2}}{\mathrm{8}} \\ $$$$\left(\mathrm{3}\right)\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\mathrm{csc}^{\mathrm{2}} \mathrm{x}\:\mathrm{dx}=−\mathrm{1}\:\:\:\left(\mathrm{4}\right)\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\:\mathrm{tan}^{\mathrm{2}} \:\mathrm{x}\:\mathrm{dx}=\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\left(\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}−\mathrm{1}\right)\mathrm{dx}=\left[\:\mathrm{tan}\:\mathrm{x}−\mathrm{x}\:\right]_{\mathrm{0}} ^{\pi/\mathrm{4}} =\frac{\mathrm{4}−\pi}{\mathrm{4}}\: \\ $$$$\mathrm{I}=\:\begin{vmatrix}{\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:}\\{\frac{\pi+\mathrm{2}}{\mathrm{8}}\:\:\:\:\:\frac{\pi+\mathrm{2}}{\mathrm{8}}\:\:\:\:\:−\mathrm{1}}\\{\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\frac{\pi+\mathrm{2}}{\mathrm{8}}\:\:\:\:\:\:\frac{\mathrm{4}−\pi}{\mathrm{4}}}\end{vmatrix} \\ $$