(a) int of -arc tan root x is(b) int of-cos^4 x is

(a)  int of: tan^-1x^1/2.dx

take tan-¹x^1/2 = t

dx/(1+x).2x^1/2 = dt

so tan t = x^1/2  or x = tan²x

⁼ int of: 2t.tan t(1+tan²t).dt

⁼ int of: 2t.tan t.sec²t.dt

integrating by parts gives :

= 2 [ t.tant²t/2 - int( (tan²t) /2 .dt) ]

= t.tan²t - int( (sec²t) -1).dt

= t.tan²t - tan t + t +C  

=x.tan^-1x^1/2 - x^1/2 + tan^-1x^1/2 + C

=(x+1)tan^-1x^1/2 - x^1/2  + C :where C is constant

(b)  int of: cos^4x.dx

Note that:   1 + cos 2A = 2cos²A

so  1 + cos 2x = 2cos²x

   =int of :  ((1 + cos 2x) / 2 )²

   = int of :  1/4*(1 + 2cos 2x+ cos²2x).dx

   = int of :  [ 1/4*(1 + 2cos 2x).dx +1/8*(1 + cos 4x).dx ]

   =x/4 + sin 2x /4 +x/8 +sin 4x /32 +C where C is constant

   =3x/8 + (sin 2x)/4 + (sin 4x)/32 +C         :where C is constant