compute the given improper integral. enter the exact value of the answer.\n int_{5}^{13} \frac{1}{sqrt{x^{2}…
compute the given improper integral. enter the exact value of the answer.\n int_{5}^{13} \frac{1}{sqrt{x^{2}-25}} dx=
Answer
Explanation:
Step1: Recall the integral formula
The integral of $\frac{1}{\sqrt{x^{2}-a^{2}}}$ is $\ln|x + \sqrt{x^{2}-a^{2}}|+C$ for $x\geq a$. Here $a = 5$.
Step2: Apply the fundamental theorem of calculus
We have $\int_{5}^{13}\frac{1}{\sqrt{x^{2}-25}}dx=\left[\ln|x+\sqrt{x^{2}-25}|\right]_{5}^{13}$.
Step3: Evaluate the definite - integral
$\ln|13+\sqrt{13^{2}-25}|-\ln|5+\sqrt{5^{2}-25}|=\ln(13 + 12)-\ln(5+0)=\ln(25)-\ln(5)$.
Step4: Use the logarithm property
Using the property $\ln m-\ln n=\ln\frac{m}{n}$, we get $\ln\frac{25}{5}=\ln5$.
Answer:
$\ln5$