Question Number 114773 by soumyasaha last updated on 21/Sep/20

     Evaluate:  ∫ (1/(sin^5 x + cos^5 x)) dx

Answered by MJS_new last updated on 21/Sep/20

∫(dx/(sin^5  x +cos^5  x))=       [t=x+(π/4) → dx=dt]  =∫(dt/(sin^5  (t+(π/4)) −cos^5  (t+(π/4))))=       [use trigonometric transformation  formulas]  =2(√2)∫(dt/((5−sin^4  t)sin t))=  =((2(√2))/5)(∫(dt/(sin t))+∫((sin t)/( (√5)−2sin^2  t))dt−∫((sin t)/( (√5)+2sin^2  t))dt)  and these are easy to solve

Commented bysoumyasaha last updated on 21/Sep/20

     It will be ∫ (dt/(sin^5 (t+(π/4))+cos^5 (t+(π/4))))   Fail to understand the negative sign.

Commented bysoumyasaha last updated on 21/Sep/20

  how +cos^5 x = −cos^5 (x−(π/4)) ?

Commented bysoumyasaha last updated on 21/Sep/20

Kindly show the inbetween steps...

Commented byMJS_new last updated on 21/Sep/20

sin (t−(π/4)) =−cos (t+(π/4))  cos (t−(π/4)) =sin (t+(π/4))  corrected again, sorry have been in a hurry  the solution is still right, it′s only that line...

Commented byMJS_new last updated on 21/Sep/20

−cos (t+(π/4)) =((√2)/2)(sin t −cos t)  sin (t+(π/4)) =((√2)/2)(sin t +cos t)  (((√2)/2)(s−c))^5 +(((√2)/2)(s+c))^5 =  =((√2)/4)s(s^4 +10s^2 c^2 +5c^4 )=       [c=(√(1−s^2 ))]  =((√2)/4)s(5−4s^4 )

Commented bysoumyasaha last updated on 21/Sep/20

Thanks Sir..