Evaluate:

Let $5 x+3=\lambda(2 x-1)+\mu$

$5 x+3=2 x \lambda-\lambda+\mu$

comparing coefficients we get

$2 \lambda=5 ;-\lambda+\mu=3$

$\Rightarrow \lambda=\frac{5}{2} ; \mu=\frac{11}{2}$

Replacing $5 x+3$ by $\lambda(2 x-1)+\mu$ in the given equation we get

$\Rightarrow \int \sqrt{2 \mathrm{x}-1} \lambda(2 \mathrm{x}-1)+\mu \mathrm{dx}$

$\Rightarrow \lambda \int(2 \mathrm{x}-1) \sqrt{2 \mathrm{x}-1} \mathrm{dx}+\int \sqrt{2 \mathrm{x}-1} \mu \mathrm{dx}$

$\Rightarrow\left(\lambda \int(2 \mathrm{x}-1)^{\frac{3}{2}} \mathrm{dx}-\mu \int(2 \mathrm{x}-1)^{\frac{1}{2}} \mathrm{dx}\right)$

$\Rightarrow \frac{5}{2} \times \frac{(2 x-1)^{\frac{5}{2}}}{2 \times \frac{5}{2}}-\frac{11}{2} \times \frac{(2 x-1)^{\frac{3}{2}}}{2 \times \frac{3}{2}}+C$

$\Rightarrow \frac{(2 x-1)^{\frac{5}{2}}}{2}-\frac{11(2 x-1)^{\frac{3}{2}}}{6}+C$