Integration by parts: Âºln(x)dx | AP Calculus BC | Khan Academy

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Worked example of finding an indefinite integral using integration by parts, where the integrand isn't a product. Created by Sal Khan.

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519, 435 students have been helped in a matter of 3.5 mins thank you

alanfaraj

Awesome, thanks Sal. I'm from Scotland studying Advanced Higher Mathematics at school, my maths teachers - despite how good they are - always said that you can't integrate ln(x). Now I know you can and how to do it! Thank you very much.

MrMal

so THATS the formula for integrating 2 functions that are multiplied! Thanks a lot this was a huge help!!!

volcanus

In under 2 minutes I finally understood where this came from!

sanveersookdawe

If you swap the f(x) and g(x), you'll get right back where you started with the original integral. Choose wisely.

rwayne

Thanks, ill start working on my calculus assignment tomorrow.
The exact equation is integral sumbol:[xln(x+1)dx]

Kameeljon

Oh lol... I'm doing calc II and realized, we keep doing exercises like d/dx of ln(x) and I'm freaking out wondering why we haven't learned to integrate it yet, thinking I missed it somehow in my log/exp chapter. Very cool to stumble onto something for down the road. Watched it anyway and it makes sense; of course, we all know the book will have much harder examples as per the uzh.

EmpyreanLightASMR

Interesting, the interval from [0, 1] where C = 0, gives an area of (-1), it converges, and [0, e] gives an area of 0, but after that it just goes to infinity. looking at and comparing the graphs of f(x) = ln(x) and F(x) = x ln(x)-x+C is interesting too.

rhoadess

you can just plug the x+1 into x or if it's easier, try doing a substitution by making x + 1 = u... and just solve f(u)du and substitute back later

not knowing what the exact equation is problematic to trying to help also

ynx

Sal you had a constant of integration from choosing g(x) that is going to affect your final answer? When you 'check' you see that constant must be zero.

djelkin

So I got a much different answer with u sub. (S means integral)
U=ln(x) du=(1/x)dx
S(ln(x)dx)=
S(x*ln(x)*(1/x)dx)=
S(e^(ln(x))*ln(x)*(1/x)dx)
Substitute u in for ln(x)
S(e^(u)*u*du)=e^u
Substitute back to ln(x)
e^ln(x)=x
What did I do wrong and how did I get the derivative with respect to y instead?

henryastor

Short: yes.
Senseless: how, exactly?

MisledTrick

You must make the formula clear like if you have one function and an integral of the derivative of another function then the formula would be completely clear.

sriramsundar