Previous Year NEET Question- Indefinite Integral

1. Answer: $\frac{x}{2}+\frac{3}{4}\ln(x+2)-\frac{1}{4}\ln(x+1)+C$

Explanation:

We can use partial fractions to decompose the integrand:

$$\frac{x^2+2x+1}{x^2+4x+3} = \frac{A}{x+1}+\frac{B}{x+3}$$

Multiplying both sides by $x^2+4x+3$, we get:

$$x^2+2x+1 = A(x+3)+B(x+1)$$

Setting $x=-1$, we get $A=1$. Setting $x=-3$, we get $B=-1$. Substituting these values back into the partial fraction decomposition, we get:

$$\frac{x^2+2x+1}{x^2+4x+3} = \frac{1}{x+1}-\frac{1}{x+3}$$

Now, we can integrate each term of the series