# Evaluate the following definite integrals. int_5^{3sqrt5}frac{dx}{sqrt{x^2-9}}

Question
Integrals
Evaluate the following definite integrals.
$$\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}$$

2021-02-26
$$\text{Given integrals is }\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}$$
$$\text{Consider x=3 sec(u)}$$
$$\text{Suvstitute x=3 sec(u) in }\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}$$
$$\int_5^{3\sqrt5}\frac{dx}{\sqrt{x^2-9}}=\int_{\sec^{-1} (\frac{5}{4})}^{\sec^{-1}(\sqrt5)} \sec(u)du$$
$$=[\ln(\tan u+\sec u)]_{\sec^{-1} (\frac{5}{4})}^{\sec^{-1}(\sqrt5)}$$
$$=\ln(\tan(\sec^{-1}(\sqrt5))+\sec(\sec^{-1}(\sqrt5)))-\ln(\tan(\sec^{-1}(\frac{5}{3}))+\sec(\sec^{-1}(\frac{5}{3})))$$
$$=\ln(2+\sqrt5)-\ln3$$
$$=0.34502$$

### Relevant Questions

Evaluate the following definite integrals.
$$\int_{1/8}^1\frac{dx}{x\sqrt{1+x^{2/3}}}$$
Evaluate each of the following integrals.
$$\int\frac{e^{x}}{1+e^{x}}dx$$
Evaluate the following integrals.
$$\int_{\ln2}^{\ln3}\frac{e^x+e^{-x}}{e^{2x}-2+e^{-2x}}dx$$
Evaluate the following integrals.
$$\int_1^{e^2}\frac{(\ln x)^5}{x}dx$$
Evaluate the following integral.
$$\int \frac{3x^{2}+\sqrt{x}}{\sqrt{x}}dx$$
Evaluate the following derivatives.
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{7}}}^{{{x}}}}\sqrt{{{1}+{t}^{{{4}}}+{t}^{{{6}}}}}{\left.{d}{t}\right.}$$
$$\int x^2 6^{x^3+8}dx$$
Evaluate the following iterated integrals. $$\int_1^3\int_1^2(y^2+y)dx\ dy$$
$$\int_1^3\int_0^{\pi/2}x\sin y\ dy\ dx$$
$$\int_{1}^{\infty}\frac{1}{x^{3}}dx$$