Question Number 177296 by peter frank last updated on 03/Oct/22

Commented by peter frank last updated on 03/Oct/22

Answered by Ar Brandon last updated on 03/Oct/22
![I=∫_0 ^π (x/(a^2 cos^2 x+b^2 sin^2 x))dx=∫_0 ^π ((π−x)/(a^2 cos^2 x+b^2 sin^2 x))dx =(π/2)∫_0 ^π (1/(a^2 cos^2 x+b^2 sin^2 x))dx=(π/2)∫_0 ^π ((sec^2 x)/(a^2 +b^2 tan^2 x))dx =π∫_0 ^(π/2) ((d(tanx))/(a^2 +b^2 tan^2 x))=(π/(ab))[arctan(((btanx)/a))]_0 ^(π/2) =(π^2 /(2ab))](https://www.tinkutara.com/question/Q177300.png)
Commented by peter frank last updated on 03/Oct/22

Answered by BaliramKumar last updated on 03/Oct/22
![I = ∫_0 ^π (x/(a^2 cos^2 x+b^2 sin^2 x))dx ..................[i] I = ∫_0 ^π (((π−x))/(a^2 cos^2 (π−x)+b^2 sin^2 (π−x)))dx I = ∫_0 ^π ((π−x)/(a^2 cos^2 x+b^2 sin^2 x))dx I = ∫_0 ^π (π/(a^2 cos^2 x+b^2 sin^2 x))dx− ∫_0 ^π (x/(a^2 cos^2 x+b^2 sin^2 x))dx I = ∫_0 ^π (π/(a^2 cos^2 x+b^2 sin^2 x))dx− I 2I = π∫_0 ^π (1/(a^2 cos^2 x+b^2 sin^2 x))dx I = (π/2)∫_0 ^π ((sec^2 x)/(a^2 +b^2 tan^2 x))dx I = (π/2)∫_0 ^π ((sec^2 (π−x))/(a^2 +b^2 tan^2 (π−x)))dx I = (π/2)∙2∫_0 ^(π/2) ((sec^2 x)/(a^2 +b^2 tan^2 x))dx I = π∫_0 ^(π/2) ((sec^2 x)/(a^2 + (btanx)^2 ))dx let b∙tan(x) = y when x=0 then y=0 sec^2 x∙dx = (dy/b) when x=(π/2) then y=∞ I = (π/b)∫_0 ^∞ (1/(a^2 +y^2 ))dy I = (π/b)∙(1/a)[tan^(−1) ((y/a))]_0 ^∞ I = (π/(ab))[(π/2)−0] I = (π^2 /(2ab))](https://www.tinkutara.com/question/Q177302.png)
Commented by peter frank last updated on 03/Oct/22
