Evaluate:

Question:

Evaluate: $\int \frac{2 x+1}{\sqrt{3 x+2}} d x$

Solution:

Let $2 x+1=\lambda(3 x+2)+\mu$

$2 x+1=3 x \lambda+2 \lambda+\mu$

comparing coefficients we get

$3 \lambda=2 ; 2 \lambda+\mu=1$

$\Rightarrow \lambda=\frac{2}{3} ; \mu=\frac{-1}{3}$

Replacing $2 x+1$ by $\lambda(3 x+2)+\mu$ in the given equation we get

$\Rightarrow \int \frac{\lambda(3 \mathrm{x}+2)+\mu}{\sqrt{3 \mathrm{x}+2}} \mathrm{dx}$

$\Rightarrow \lambda \int \frac{3 x+2}{\sqrt{3 x+2}} d x+\mu \int \frac{1}{\sqrt{3 x+2}} d x$

$\Rightarrow\left(\lambda \int \sqrt{3 x+2} d x-\mu \int(3 x+2)^{\frac{-1}{2}} d x\right)$

$\Rightarrow \frac{2}{3} \times \frac{(3 x+2)^{\frac{3}{2}}}{3 \times \frac{3}{2}}-\frac{1}{3} \times \frac{(3 x+2)^{\frac{1}{2}}}{3 \times \frac{1}{2}}+c$

$\Rightarrow \frac{4(3 x+2)^{\frac{3}{2}}}{27}-\frac{2(3 x+2)^{\frac{1}{2}}}{9}+c$

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