∫20dxx+4−x2=∫20dx−(x2−x−4)
=∫20dx−(x2−x+14−14−4)
=∫20dx−[(x−12)2−174]
=∫20dx(√172)2−(x−12)2
Let x−12=t⇒dx=dt
When x=0,t=−12 and when x=2,t=32
∴∫20dx(√172)2−(x−12)2=∫32−12dt(√172)2−t2
=⎡⎢
⎢⎣12(√172)log√172+t√172−t⎤⎥
⎥⎦32−12
=1√17⎡⎢⎣log√172+32√172−32−log√172−12√172+12⎤⎥⎦
=1√17[log√17+3√17−3−log√17−1√17+1]
=1√17[log√17+3√17−3×√17+1√17−1]
=1√17log[17+3+4√1717+3−4√17]
=1√17log[20+4√1720−4√17]
=1√17log(5+√175−√17)
=1√17log[(5+√17)(5+√17)25−17]
=1√17log[25+17+10√178]
=1√17log(42+10√178)
=1√17log(21+5√174)