Q.
∫x+2x2+2x+3dx is equal to
see full answer
Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!
Real Strategies. Real People. Real Success Stories - Just 1 call away
An Intiative by Sri Chaitanya
a
x2+2x+3+logx+1+x2+2x+3+C
b
x2+2x+3−logx+1+x2+2x+3+C
c
x2+2x+3+logx+1−x2+2x+3+C
d
None of the above
answer is A.
(Unlock A.I Detailed Solution for FREE)
Ready to Test Your Skills?
Check your Performance Today with our Free Mock Test used by Toppers!
Take Free Test
Detailed Solution
Let x+2=Addxx2+2x+3+B ⇒ x+2=A(2x+2)+B⇒ x+2=2Ax+(2A+B)On equating the coefficient of x and constant term on both sides, we get 2A=1⇒A=12and 2A+B=2⇒2×12+B=2⇒ B=2-1=1 ⇒ x+2=122x+2+1∴∫x+2x2+2x+3dx=∫12(2x+2)+1x2+2x+3dx =12∫2x+2x2+2x+3dx+∫dxx2+2x+3Let l1=∫2x+2x2+2x+3dx and I2=∫dxx2+2x+3Then,∫x+2x2+2x+3dx=12I1+I2 .....(i)Now, I1=∫2x+2x2+2x+3dxLet x2+2x+3=t⇒(2x+2)dx=dt I2=∫dtt=∫t−1/2dt=t(−1/2)+11/2+1+C=2t+C1 =2x2+2x+3+C1and I2=∫dxx2+2x+3=∫dxx2+2x+(1)2+3−(1)2 =∫dx(x+1)2+(2)2Let x+1=t⇒dx=dtI2=∫dtt2+(2)2=logt+t2+2+C2 ∵∫dxx2+a2=log∣x+x2+a2=logx+1+(x+1)2+2+C2=logx+1+x2+2x+3+C2On putting the values of l1 and l2 in Eq. (i), we get∫x+2x2+2x+3dx=122x2+2x+3+logx+1+x2+2x+3∵12C1+C2=C=x2+2x+3+log∣x+1+x2+2x+3+C
Watch 3-min video & get full concept clarity