Integration Question

[Ben Z]

Does anyone know how to integrate:



alternatively:



Where x is a variable and a is a constant. What formula I could use?

Thank you very much.

Ben Z.

 

[LCKurtz]

I would factor the [MATH]x^3-a[/MATH] into a linear factor and an irreducible quadratic and use partial fractions.

 

[lookagain]

Does anyone know how to integrate:

View attachment 17790

alternatively:

View attachment 17791

Where x is a variable and a is a constant. What formula I could use?

Thank you very much.

Ben Z.

Ben Z., those are not equivalent to each other. Look what happens if
a = 0 and x = 8.

Suppose a + x = y^3

Then a new expression could be:

\(\displaystyle \dfrac{y}{y^3 - a}\)

 

[LCKurtz]

Ben Z., those are not equivalent to each other.

He didn't say they were equivalent. The substitution [MATH]u=\sqrt[3]{a+x}[/MATH] leads to a simple term plus a term [MATH]\frac 1 {u^3-a}[/MATH], so solving that is indeed a strategy for being able to solve the original.