$\int 1/(x^3-25x) \mathrm{d}x$

Question

Integrate $\displaystyle \frac{1}{x^3-25x}$

Between limits of $4,3$.

Think it involves the 'Difference in squares' substitution.

I ended up with an answer of $0.3242$ which is wrong as it is $0.023$.

Any help and pointers would be greatly appreciated.

Matt

Answer 1

Hint:

$$\frac{1}{x^3-25x} = \frac{1}{x(x+5)(x-5)} = \frac{A}{x} + \frac{B}{x+5} + \frac{C}{x-5}.$$

Solve for $A,B,C$ using partial fractions (laid out above), and then observe that the integral of a sum is the sum of the integrals.

Answer 2

John's approach is correct. $\frac{1}{x^3-25x} = \frac{1}{x(x+5)(x-5)} = \frac{A}{x} + \frac{B}{x+5} + \frac{C}{x-5}$.

Continuing, we generate the following equation: $$A(x+5)(x-5)+Bx(x-5)+Cx(x+5)=1$$

By strategically choosing our $x$ values (I recommend $x=-5,0,$ and $5$), we can derive that $A=\frac {-1}{25}, B= \frac{1}{50},$ and $C= \frac {1}{50}$. Thus, we rewrite:

$\int \frac {1}{x^3-25x}= \int \frac{-1/25}{x} + \frac{1/50}{x+5} + \frac{1/50}{x-5} \mathrm{d} x$.

I suggest breaking the integral into three separate integrals and proceed by setting the denominator of each equal to some dummy variable. Then, all three very nicely fit the form $\int \frac {\mathrm{d} u} {u}$. Can you proceed from here?

Answer 3

HINT:

$$\int \frac{dx}{x^3-25x}=\frac{1}{50}\int \frac{\frac{50dx}{x^3}}{1-\frac{25}{x^2}}$$

Now put $${1-\frac{25}{x^2}}=t$$