Evaluate:

Question:

Evaluate: $\int \frac{3 x+5}{\sqrt{7 x+9}} d x$

Solution:

Let $3 x+5=\lambda(7 x+9)+\mu$

$3 x+5=7 x \lambda+9 \lambda+\mu$

comparing coefficients, we get

$7 \lambda=3 ; 9 \lambda+\mu=1$

$\Rightarrow \lambda=\frac{3}{7} ; \mu=\frac{8}{7}$

Replacing $3 x+5$ by $\lambda(7 x+9)+\mu$ in the given equation we get

$\Rightarrow \int \frac{\lambda(7 \mathrm{x}+9)+\mu}{\sqrt{7 \mathrm{x}+9}} \mathrm{dx}$

$\Rightarrow \lambda \int \frac{7 \mathrm{x}+9}{\sqrt{7 \mathrm{x}+9}} \mathrm{dx}+\mu \int \frac{1}{\sqrt{7 \mathrm{x}+9}} \mathrm{dx}$

$\Rightarrow\left(\lambda \int \sqrt{7 \mathrm{x}+9} \mathrm{dx}+\mu \int(7 \mathrm{x}+9)^{\frac{-1}{2}} \mathrm{dx}\right)$

$\Rightarrow \frac{3}{7} \times \frac{(7 x+9)^{\frac{3}{2}}}{7 \times \frac{3}{2}}+\frac{8}{7} \times \frac{(7 x+9)^{\frac{1}{2}}}{7 \times \frac{1}{2}}+c$

$\Rightarrow \frac{6(7 x+9)^{\frac{3}{2}}}{147}-\frac{16(7 x+9)^{\frac{1}{2}}}{49}+C$

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