Let 2x+3=Mddxx2+4x+3+N⇒ 2x+3=M(2x+4)+NEquating the coefficients of x and constant terms on both sides, we get 2=2M⇒M=1 and 3=4M+N⇒ N=3−4×1=−1∴ 2x+3=(2x+4)−1Hence I=∫[(2x+4)−1]x2+4x+3dx=∫(2x+4)x2+4x+3dx−∫x2+4x+3dx=I1−I2, (say) Now, I1=∫(2x+4)x2+4x+3dxPutting x2+4x+3=t⇒ (2x+4)dx=dt, we have I1=∫t1/2dt=t3/23/2+c1=23x2+4x+33/2+c1I2=∫x2+4x+3dx=∫(x+2)2−12dx=12(x+2)(x+2)2−12−12⋅12log⁡x+2+(x+2)2−12+c2=12(x+2)x2+4x+3−12log⁡x+2+x2+4x+3+c2∴ I=I1−I2=23x2+4x+33/2−12(x+2)x2+4x+3⇒I=−12log⁡x+2+x2+4x+3+c, where ,c=c1−c2,

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$\int (2 x + 3) \sqrt{x^{2} + 4 x + 3} d x =$

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$- \frac{1}{2} \log {|x + 2 + \sqrt{x^{2} + 4 x + 3}|}_{1}] + c$

$\log |(x + 2) + (\sqrt{x^{2} + 4 x + 3})| + c$

$\log |(x + 2) + (\sqrt{x^{2} + 4 x + 3})| - c$

$\log |(x - 2) + (\sqrt{x^{2} + 4 x + 3})| + c$

answer is D.

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Detailed Solution

Let $2 x + 3 = M \frac{d}{d x} (x^{2} + 4 x + 3) + N$

$\Rightarrow 2 x + 3 = M (2 x + 4) + N$

Equating the coefficients of x and constant terms on both sides, we get $2 = 2 M \Rightarrow M = 1 and 3 = 4 M + N$

$\begin{array}{ll} \Rightarrow N = 3 - 4 \times 1 = - 1 \\ ∴ 2 x + 3 = (2 x + 4) - 1 \end{array}$

Hence $I = \int [(2 x + 4) - 1] \sqrt{x^{2} + 4 x + 3 d x}$

$\begin{array}{l} = \int (2 x + 4) \sqrt{x^{2} + 4 x + 3} d x - \int \sqrt{x^{2} + 4 x + 3} d x \\ = I_{1} - I_{2}, (say) \end{array}$

Now, $I_{1} = \int (2 x + 4) \sqrt{x^{2} + 4 x + 3} d x$

Putting $x^{2} + 4 x + 3 = t$

$\Rightarrow (2 x + 4) d x = d t, we have I_{1} = \int t^{1 / 2} d t = \frac{t^{3 / 2}}{3 / 2} + c_{1} = \frac{2}{3} {(x^{2} + 4 x + 3)}^{3 / 2} + c_{1}$

$\begin{array}{l} I_{2} = \int \sqrt{x^{2} + 4 x + 3} d x = \int \sqrt{(x + 2)^{2} - 1^{2}} d x \\ = \frac{1}{2} (x + 2) \sqrt{(x + 2)^{2} - 1^{2}} - \frac{1}{2} \cdot 1^{2} \log |x + 2 + \sqrt{(x + 2)^{2} - 1^{2}}| + c_{2} \\ = \frac{1}{2} (x + 2) \sqrt{x^{2} + 4 x + 3} - \frac{1}{2} \log |x + 2 + \sqrt{x^{2} + 4 x + 3}| + c_{2} \end{array}$

$\begin{array}{l} ∴ I = I_{1} - I_{2} = \frac{2}{3} {(x^{2} + 4 x + 3)}^{3 / 2} - [\frac{1}{2} (x + 2) \sqrt{x^{2} + 4 x + 3} \Rightarrow I = - \frac{1}{2} \log |x + 2 + \sqrt{x^{2} + 4 x + 3}|] + c \\ , (where, c = c_{1} - c_{2}), \end{array}$