How can I integrate t^2 * ln(t) using the integration by parts method?

[Bikkehaug]

Homework Statement

I'm not sure if this is pre calculus or calculus and beyond. If I've posted in the wrong section, please feel free to move this topic.

∫t² * lnt dt

Homework Equations

The Attempt at a Solution

I've tried using the integration by parts:

∫uv' dt = uv - ∫u'v dt

u = lnt u' = 1/t

v = t³ / 3 v' = t²

I then get the following:

lnt * t³ / 3 - ∫1/t * t³ / 3

I'm not sure where to continue from this.

 

[Ray Vickson]

Homework Statement

I'm not sure if this is pre calculus or calculus and beyond. If I've posted in the wrong section, please feel free to move this topic.

∫t² * lnt dt

Homework Equations

The Attempt at a Solution

I've tried using the integration by parts:

∫uv' dt = uv - ∫u'v dt

u = lnt u' = 1/t

v = t³ / 3 v' = t²

I then get the following:

lnt * t³ / 3 - ∫1/t * t³ / 3

I'm not sure where to continue from this.

What is preventing you from integrating $(1/t)(t^3/3) = t^2 /3$?

 

[Bikkehaug]

What is preventing you from integrating $(1/t)(t^3/3) = t^2 /3$?

I thought I needed to do partial integration again for that one, since it's another set of products?

 

[PeroK]

I then get the following:

lnt * t³ / 3 - ∫1/t * t³ / 3

I'm not sure where to continue from this.

You've done the hard bit!

 

[micromass]

I thought I needed to do partial integration again for that one, since it's another set of products?

The integral

$$\int x^n dx$$

is a special case and you should have it memorized.

 

[PeroK]

I thought I needed to do partial integration again for that one, since it's another set of products?

You are allowed to simplify things before integrating them.

 

[Mark44]

I'm not sure if this is pre calculus or calculus and beyond.

Problems involving integration or differentiation definitely should be posted in Calculus & Beyond, so this is the right forum section.