The correct option is B 16(2x2+3)32+x2√2x2+3+32√2ln∣∣∣x+√x2+32∣∣∣+C
I=∫(x+1)√2x2+3dx
Let x+1=λddx(2x2+3)+μ
⇒x+1=4xλ+μ
Equating coefficient of same degree term, we get
⇒λ=14,μ=1⇒I=∫(14(4x)+1)√2x2+3dx⇒I=14∫4x√2x2+3dx+∫√2x2+3dx
Putting 2x2+3=t⇒4x dx=dt
Now,
⇒I=14∫√t dt+∫√2x2+3dx =t326+∫
⎷2[x2+(√32)2]dx⇒I=16(2x2+3)32+√2∫√x2+(√32)2dx∴I=16(2x2+3)32+x2√2x2+3+32√2ln∣∣∣x+√x2+32∣∣∣+C