∫101√1+x+√xdx
u+v=√x
⇒dx=2√xdu
=2∫u√u2+u+1du
=∫(2u+12√u2+u+1−12√u2+u+1)du
For ∫2u+12√u2+u+1du
u+v=u2+v+1
⇒du=12u+1dv
=∫1√vdv
=2√v
=2√v2+v+1
For =∫1√u2+u+1du
=−∫1√(1+12)2+34du
u+v=2v+1√3
⇒du=√32dv
=∫1√v2+1dv
=ln(√v2+1+v)
=ln⎛⎜⎝
⎷(2v+1√3)2+1+2v+1√3⎞⎟⎠
So integral is
2√x+√x+1=ln(2√x+√x+1+√x)+1)+c
Putting the values
=0.6964.
![1213323_1319966_ans_ffe4e884b3b240f3a24329fffb2c7dcf.jpg](https://search-static.byjusweb.com/question-images/toppr_invalid/questions/1213323_1319966_ans_ffe4e884b3b240f3a24329fffb2c7dcf.jpg)