Menu Close

Evaluate-0-pi-x-a-2-cos-2-x-b-2-sin-2-x-dx-




Question Number 177296 by peter frank last updated on 03/Oct/22
  Evaluate   ∫_0 ^π (x/(a^2 cos^2 x+b^2 sin^2 x))dx
Evaluate0πxa2cos2x+b2sin2xdx
Commented by peter frank last updated on 03/Oct/22
ans      (π^2 /(2ab))
ansπ22ab
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))
I=0πxa2cos2x+b2sin2xdx=0ππxa2cos2x+b2sin2xdx=π20π1a2cos2x+b2sin2xdx=π20πsec2xa2+b2tan2xdx=π0π2d(tanx)a2+b2tan2x=πab[arctan(btanxa)]0π2=π22ab
Commented by peter frank last updated on 03/Oct/22
thank you
thankyou
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))
I=0πxa2cos2x+b2sin2xdx[i]I=0π(πx)a2cos2(πx)+b2sin2(πx)dxI=0ππxa2cos2x+b2sin2xdxI=0ππa2cos2x+b2sin2xdx0πxa2cos2x+b2sin2xdxI=0ππa2cos2x+b2sin2xdxI2I=π0π1a2cos2x+b2sin2xdxI=π20πsec2xa2+b2tan2xdxI=π20πsec2(πx)a2+b2tan2(πx)dxI=π220π2sec2xa2+b2tan2xdxI=π0π2sec2xa2+(btanx)2dxletbtan(x)=ywhenx=0theny=0sec2xdx=dybwhenx=π2theny=I=πb01a2+y2dyI=πb1a[tan1(ya)]0I=πab[π20]I=π22ab
Commented by peter frank last updated on 03/Oct/22
thank you
thankyou

Leave a Reply

Your email address will not be published. Required fields are marked *