Partial Fractions

[Firepanda]

Integrate using partial fractions:

(int) (x^3)/(x^2 -1) dx

I have put into the form (int) (x^3)/((x-1)(x+1)) dx

I thought partial fractions had this property:

'Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.'

And this obviously doesn't, so how do you do it?

 

[HallsofIvy]

Integrate using partial fractions:

(int) (x^3)/(x^2 -1) dx

I have put into the form (int) (x^3)/((x-1)(x+1)) dx

I thought partial fractions had this property:

'Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.'

And this obviously doesn't, so how do you do it?

Divide! What is x³ divided by x²- 1? What are the quotient and remainder?

 

[Firepanda]

ahhh

so it =

x + 1/(x-1) - 1/(x+1)(x-1)

:D

 

[Firepanda]

While I am here I have another question:

(int) 1 to e (ln(x))/x^2 dx

THe question says, use a suitable substitution to evaluate the definite integral, I can do it by parts but I don't want to lose marks for it.

Can any1 suggest the substitution to get me started?
Thx

 

[rocomath]

While I am here I have another question:

(int) 1 to e (ln(x))/x^2 dx

THe question says, use a suitable substitution to evaluate the definite integral, I can do it by parts but I don't want to lose marks for it.

Can any1 suggest the substitution to get me started?
Thx

This should be in the Calculus section.

Verifying your problem ...

$$\int_1^e\frac{\ln x}{x^2}dx$$

 

[Firepanda]

This should be in the Calculus section.

Verifying your problem ...

$$\int_1^e\frac{\ln x}{x^2}dx$$

Yep that's the one

 

[rocomath]

Yep that's the one

Substitutions ...

$$t=\ln x \rightarrow e^t=x$$

$$dt=\frac 1 x dx$$

You will need to do parts afterwards.