\(\int\frac{3x+4}{\sqrt{5x^2+8}}dx\)
= \(\int(\frac{3x}{\sqrt{5x^2+8}}+\frac4{\sqrt{5x^2+8}})dx\)
= \(\frac3{10}\int\frac{10x}{\sqrt{5x^2+8}}dx+\frac4{\sqrt 5}\int\frac{dx}{\sqrt{x^2+(\sqrt{\frac85})^2}}\)
= \(\frac3{10}(2\sqrt{5x^2+8})\) + \(\frac4{\sqrt 5}log|x + \sqrt{x^2+\frac85}|+k\)
= \(\frac35\sqrt{{5x^2+8}}\) + \(\frac4{\sqrt 5}log(\frac{\sqrt5x+\sqrt{5x^2+8}}{\sqrt5})+k\)
= \(\frac35\sqrt{{5x^2+8}}\) + \(\frac4{\sqrt 5}log(\sqrt 5x+\sqrt{5x^2+8})\) - \(\frac4{\sqrt5}log\sqrt5+k\)
= \(\frac35\sqrt{{5x^2+8}}\) + \(\frac4{\sqrt 5}log(\sqrt5x + \sqrt{5x^2+8})+c\)
