Question:
Evaluate: $\int \frac{1}{(7 x-5)^{3}}+\frac{1}{\sqrt{5 x-4}} d x$
Solution:
Let $I=\int \frac{1}{(7 x-5)^{2}}+\frac{1}{\sqrt{5 x-4}} \mathrm{dx}$ then,
$I=\int(7 x-5)^{-3}+(5 x-4)^{-\frac{1}{2}}$
$=\frac{(7 x-5)^{-3+1}}{7(-3+1)}+\frac{(5 x-4)^{\frac{-1}{2}+1}}{5\left(-\frac{1}{2}+1\right)}$
$=\frac{(7 x-5)^{-2}}{-14}+\frac{(5 x-4)^{\frac{1}{2}}}{5\left(\frac{1}{2}\right)}$
Hence, $I=-\frac{1}{14}(7 x-5)^{-2}+\frac{2}{5} \sqrt{5 x-4}+C$
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