Use a table of integrals to evaluate the integral. $\int \frac{\sqrt{4 x+5}}{x^{2}} d x$
A) $-\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$
B) $-\frac{\sqrt{4 x+5}}{x}+\frac{2 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$
C) $-\frac{\sqrt{4 x+5}}{x}+\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$
D) $-\frac{4 \sqrt{5}}{5} \ln \left|\frac{\sqrt{4 x+5}-\sqrt{5}}{\sqrt{4 x+5}+\sqrt{5}}\right|+C$

Question

Quizplus · Accepted Answer

Using the substitution $u=\sqrt{4x+5}$, we get $du=\frac{2}{\sqrt{4x+5}}\,dx$ and $x=\frac{1}{4}(u^2-5)$. Substituting these in the given integral, we get $$\int\frac{\sqrt{4x+5}}{x^2}\,dx=\int\frac{u}{\frac{1}{16}(u^2-5)^2}\cdot\frac{2}{u}\,du=-\frac{1}{4}\int\frac{du}{(u^2-5)^{\frac{3}{2}}}.$$ Using the formula $\int\frac{du}{(u^2-a^2)^{\frac{3}{2}}}=-\frac{1}{a^2\sqrt{u^2-a^2}}+C$, we get $$\int\frac{\sqrt{4x+5}}{x^2}\,dx=-\frac{1}{4}\cdot\frac{1}{\sqrt{5}\cdot\left|\sqrt{4x+5}-\sqrt{5}\right|}+C.$$ Simplifying this expression using the properties of logarithms, we can find that the answer is B.

Use a Table of Integrals to Evaluate the Integral ∫4x+5x2dx\int \frac{\sqrt{4 x+5}}{x^{2}} d x∫x24x+5​​dx

Use a Table of Integrals to Evaluate the Integral $\int \frac{\sqrt{4 x+5}}{x^{2}} d x$