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Question Number 163158 by mnjuly1970 last updated on 04/Jan/22
       prove that  ∫_0 ^( (π/4)) (( sin(x)+cos(x))/( (√(1+sin(x)cos(x))))) dx= (√2) .cot^( −1) ((√2) )     −−−−−
provethat0π4sin(x)+cos(x)1+sin(x)cos(x)dx=2.cot1(2)
Answered by mahdipoor last updated on 04/Jan/22
get sinx−cosx=(√3)u         { (((√3)du=(cosx+sinx)dx)),((3u^2 =sin^2 x+cos^2 x−2sinx.cosx⇒((1−3u^2 )/2)=sinx.cosx)) :}  ⇒∫_0 ^( π/4) ((sinx+cosx)/( (√(1+sinx.cosx))))dx=∫_(−1/(√3)) ^( 0) (((√3)du)/( (√(1+((1−3u^2 )/2)))))=  (√2)∫_( −1/(√3)) ^( 0) (du/( (√(1−u^2 ))))=[(√2)sin^(−1) u+C]_(−1/(√3)) ^0    = −(√2)sin^(−1) (((−1)/( (√3))))=(√2)sin^(−1) ((1/( (√3))))  if sinx=(1/( (√3))) ⇒ cotx=(√2)  ⇒(√2)sin^(−1) ((1/( (√3))))=(√2)cot^(−1) ((√2))
getsinxcosx=3u{3du=(cosx+sinx)dx3u2=sin2x+cos2x2sinx.cosx13u22=sinx.cosx0π/4sinx+cosx1+sinx.cosxdx=1/303du1+13u22=21/30du1u2=[2sin1u+C]1/30=2sin1(13)=2sin1(13)ifsinx=13cotx=22sin1(13)=2cot1(2)
Commented by mnjuly1970 last updated on 04/Jan/22
bravo sir mahdipoor
bravosirmahdipoor

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