∫dx13+3cosx+4sinx
=∫dx13+3⎛⎜
⎜⎝1−tan2x21+tan2x2⎞⎟
⎟⎠+4⎛⎜
⎜⎝2tanx21+tan2x2⎞⎟
⎟⎠
=(1+tan2x2)dx13+13tan2x2+3−3tan2x2+8tanx2
=sec2x2dx16+10tan2x2+8tanx2
Let t=tanx2⇒dt=12sec2x2dx
⇒2dt=sec2x2dx
=∫2dt16+10t2+8t
=∫dt8+5t2+4t
=∫dt5(t2+4t5+85)
=15∫dtt2+2×2t5+(25)2−(25)2+85
=15∫dt(t+25)2−425+85
=15∫dt(t+25)2+−4+4025
=15∫dt(t+25)2+3625
=15∫dtt2+2×2t5+(25)2−(25)2+85
=15∫dt(t+25)2−425+85
=15∫dt(t+25)2+−4+4025
=15∫dt(t+25)2+3625
=15×56tan−1⎛⎜
⎜
⎜⎝t+2565⎞⎟
⎟
⎟⎠
=16tan−1(5t+26)+c
=16tan−1⎛⎜
⎜⎝5tanx2+26⎞⎟
⎟⎠+c where t=tanx2