Weekly Math Problem #62

Improper Integral. Ever so often, I find my way back to calculus, and this week it's via an improper integral. By my recollection, the technique is okay. The type of function you're dealing with can definitely be the cause that the problem is annoying to do. Let's just hope this week's problem isn't annoying to you. 😬 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to solve an improper integral
        ✔️     Methods of integration

     

WMP! #62 says to...

Happy solving!

Check back on Saturday, November 6th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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Alright...let's get into this problem! If you looked up integral tables and used a calculator, that's good. I did too! I wasn't expecting anyone to have the information needed to solve this problem memorized. I didn't have it memorized, but I felt confident that I knew how to use the assistance.

I started this problem by using an integral table. Why? Well, I saw that the function was rational and knew that most, if not all, integral tables have some rational functions on them. When I saw what I needed, I rewrote the function in a way that help me begin solving the problem.

With the problem rewritten, I could see that I needed to invoke the substitution method to keep going. Since the substitution method uses a new variable, I also adjusted the lower bound of integration to match the new variable.

Once that was done, I could now move forward and integrate.

Since this is a definite integral, we need to evaluate! Evaluating the second term, which comes from the lower bound was simple. If you're familiar with this special value, then you can simple write the value. You can also use a calculator like I did. 😁 It's the first term, that requires a bit more work. Infinity isn't a value to plug in, so I used my calculator, again, to see if the function converges as my variable approaches infinity. It does!!!

Now that I have the values I need, I can plug them in and simplify.

I don't want to leave my answer like that, so I did a little "clean-up" by rationalizing the denominator.

Here is an image of the function. Keep in mind that the variable is approaching infinity. Even though, the shape doesn't end, the area under the curve converges to value above. 

**This plot was created using Geogebra's Graphing Calculator. Click image to enlarge.

▪️ Were you able to solve the improper integral??

▪️ Let me know what you thought about this week's problem in the comments section. 

Thanks for solving with me this week!

WMP! #63. is up next. 👩🏿‍🏫

Cheers!

The Younge Lady