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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To tackle the integral \(\int \frac{5 d y}{y \sqrt{7+(\ln y)^{2}}}\), a clever substitution is key! You can let \(u = \ln y\), which will simplify it down to \(\int \frac{5 du}{e^u \sqrt{7 + u^2}}\). This neatly converts the integral into one that table resources can handle, turning a complex task into a charmingly simple one! Once you're in the groove with the substitution, recall that the integral \(\int \frac{du}{\sqrt{a^2 + u^2}} = \ln |u + \sqrt{a^2 + u^2}| + C\). Using this formula, and plugging back \(u = \ln y\) with \(a^2 = 7\) gives you the finished product. Through this process, you'll discover that the value of the integral matches answer choice C: \(5 \ln \left|\ln y+\sqrt{7+(\ln y)^{2}}\right| + C\). Happy integrating!
