ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Proof of the integral of lnx and other examples involving the integration of natural logs.

Updated on June 25, 2012


In this hub I will show how to integrate lnx and other functions involving lnx. The best way to integrate lnx is to use integration by parts.

That is the ∫uv’ dx= uv - ∫u’vdx

If you haven’t used integration parts before then you will need to practice this before you attempt the integral of lnx.

Example 1

Work out ∫lnxdx

First rewrite this as ∫lnx.1dx

So u = lnx and v’ = 1. The reason you make u = lnx instead of u = 1 is that you should know the differential of lnx is 1/x. Therefore u’ = 1/x. Also if you integrate 1 you get x so v = x.

Next substitute u = lnx, u’ = 1/x, v = x and v’ = 1 into the above formula for integration by parts:

∫uv’ dx= uv - ∫u’vdx

∫lnx.1dx = lnx.x - ∫(1/x).xdx

Next tidy this up and (1/x) × x will cancel to give 1:

=xlnx - ∫1.dx

Finally, work out the integral of 1.dx

= xlnx – x + c

Example 2

Work out the integral of 2xln5x.

Like example 1 make u = ln5x and v’ = 2x. Always make u equal to the function with natural log.

Now differentiate u to get u’ = 1/x and integrate v’ to get v = x²

Next substitute these values into the formula for integration by parts.

∫uv’ dx= uv - ∫u’vdx

∫2x.ln5xdx = ln5x.x² - ∫(1/x).x²dx

Tidy this up and (1/x) × x² will simplify to x:

=x²ln5x - ∫x.dx

= xlnx – 0.5x² + c

Let’s look at one last example that involves the integration of natural log x.

Example 3

Work out the integral of x³ln8x,

Like example 1 and 2 use the formula for integration by parts:

Just like examples 1 and 2, make u = ln8x and v’ = x³

Now differentiate u to give u’ = 1/x and integrate v’ to give v = ¼x⁴

Next substitute these values into the formula for integration by parts.

∫uv’ dx= uv - ∫u’vdx

∫x³. ln8xdx = ln8x. ¼x⁴ - ∫(1/x).0.25x⁴dx

Tidy this up and (1/x).0.25x⁴ will simplify to get ¼x³

= ln8x. ¼x⁴ - ∫¼x³dx

= ¼x⁴ln8x – (1/16)x⁴ + c

So that is all you need to know about integration involving natural logs.

Comments

Submit a Comment

  • rahul0324 profile image

    Jessee R 

    6 years ago from Gurgaon, India

    Calculus is divine... It would be impossible to understand the dimensions of the universe without it...

    If Universe is the art.... Calculus is its describing literature..

    A very interesting hub.... well explained easy to comprehend explanations..

    Great

working

This website uses cookies

As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://hubpages.com/privacy-policy#gdpr

Show Details
Necessary
HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
LoginThis is necessary to sign in to the HubPages Service.
Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
AkismetThis is used to detect comment spam. (Privacy Policy)
HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
Features
Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
MavenThis supports the Maven widget and search functionality. (Privacy Policy)
Marketing
Google AdSenseThis is an ad network. (Privacy Policy)
Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
Index ExchangeThis is an ad network. (Privacy Policy)
SovrnThis is an ad network. (Privacy Policy)
Facebook AdsThis is an ad network. (Privacy Policy)
Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
AppNexusThis is an ad network. (Privacy Policy)
OpenxThis is an ad network. (Privacy Policy)
Rubicon ProjectThis is an ad network. (Privacy Policy)
TripleLiftThis is an ad network. (Privacy Policy)
Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
Statistics
Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)