Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{6 x-5}{\sqrt{3 x^{2}-5 x+1}} d x$

Solution:

Given $I=\int \frac{6 x-5}{\sqrt{3 x^{2}-5 x+1}} d x$

Integral is of form $\int \frac{\mathrm{px}+\mathrm{q}}{\sqrt{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}}} \mathrm{dx}$

Writing numerator as $\mathrm{px}+\mathrm{q}=\lambda\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right\}+\mu$

$\Rightarrow p x+q=\lambda(2 a x+b)+\mu$

$\Rightarrow 6 x-5=\lambda(6 x-5)+\mu$

$\therefore \lambda=1$ and $\mu=0$

Let $u=3 x^{2}-5 x+1 \rightarrow d x=\frac{1}{6 x-5} d u$

$\Rightarrow \int \frac{6 x-5}{\sqrt{3 x^{2}-5 x+1}} d x=\int \frac{1}{\sqrt{u}} d u$

We know that $\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$

$=2 \sqrt{3 x^{2}-5 x+1}+c$

$\therefore I=\int \frac{6 x-5}{\sqrt{3 x^{2}-5 x+1}} d x=2 \sqrt{3 x^{2}-5 x+1}+c$

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