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..