Evaluate:

Question:

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

Solution:

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

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

comparing coefficients we get

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

$\Rightarrow \lambda=-1 ; \mu=3$

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

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

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

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

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

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

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