9:29 PM Mon 17 Mar 90\% mylab.pearson.com Kurulus Osm... Can't Open P... CUNY Login The Evolutio... Discord Do Homewor... MAT 220-S25 Rasheed Larry 03/17/25 9:29 PM HW Score: 66.67\%, 2 Homework: Integration using Tables Question 3, 8.6.35 of 3 points Points: 0 of 1 Save Question list Question 1 Question 2 Question 3 Use a substitution to change the following integral into one that can be found in the table of integrals. Then evaluate the integral. $$ \int \frac{5 d y}{y \sqrt{7+(\ln y)^{2}}} $$ Which of the following is the value of the given integral? A. $$ 5 \ln \left|\ln y-\sqrt{7+(\ln y)^{2}}\right|+C $$ B. $$ \ln \left|\ln y+\sqrt{7-(\ln y)^{2}}\right|+C $$ C. $$ 5 \ln \left|\ln y+\sqrt{7+(\ln y)^{2}}\right|+C $$ D. $$ \ln \left|\ln y-\sqrt{7-(\ln y)^{2}}\right|+C $$ Textbook Calculator Clear all Check answer

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UpStudy AI · Accepted Answer

Let $$ u = \ln y $$, so that  
$$
du = \frac{dy}{y}.
$$
Thus, the integral becomes  
$$
\int \frac{5\,dy}{y\sqrt{7+(\ln y)^2}} = 5 \int \frac{du}{\sqrt{7+u^2}}.
$$

Recall the standard integral formula:  
$$
\int \frac{du}{\sqrt{a^2+u^2}} = \ln\left|u+\sqrt{a^2+u^2}\right|+C.
$$
Here, $$ a^2=7 $$, so the formula applies with $$ a=\sqrt{7} $$. Therefore,  
$$
5 \int \frac{du}{\sqrt{7+u^2}} = 5 \ln\left|u+\sqrt{7+u^2}\right|+C.
$$

Substituting back $$ u = \ln y $$ gives  
$$
5 \ln\left|\ln y+\sqrt{7+(\ln y)^2}\right|+C.
$$

Thus, the correct answer is  
$$
\textbf{(C)}\quad 5 \ln \left|\ln y+\sqrt{7+(\ln y)^{2}}\right|+C.
$$
The integral evaluates to $$ 5 \ln \left|\ln y+\sqrt{7+(\ln y)^{2}}\right|+C $$, so the correct choice is **C**.

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