Question

Evaluate the integral: \[ \int \frac{x}{x^2-5x+4}\,dx \]

• A. \(\displaystyle \frac{1}{3}\log\left|\frac{(x-4)^4}{x-1}\right|+c\)
• B. \(\displaystyle \frac{4}{3}\log\left|\frac{x-4}{(x-1)^4}\right|+c\)
• C. \(\displaystyle \frac{1}{3}\log\left|\frac{(x-4)^2}{x-1}\right|+c\)
• D. \(\displaystyle \frac{4}{3}\log\left|\frac{x-4}{(x-1)^4}\right|+c\)

Hint:

Whenever the denominator factorizes into distinct linear factors, partial fractions is usually the fastest integration technique.

Answer 1

The Correct Option is A

Step 1: Factor the denominator.
$x^2-5x+4=(x-1)(x-4)$, so the integrand is $\dfrac{x}{(x-1)(x-4)}$.
Step 2: Split into partial fractions.
Write $\dfrac{x}{(x-1)(x-4)}=\dfrac{A}{x-1}+\dfrac{B}{x-4}$, so $x=A(x-4)+B(x-1)$.
Step 3: Find $A$ and $B$.
Comparing: $A+B=1$ and $4A+B=0$. From these, $A=-\dfrac13$ and $B=\dfrac43$.
Step 4: Integrate each piece.
\[ \int\frac{x}{x^2-5x+4}dx=-\frac13\int\frac{dx}{x-1}+\frac43\int\frac{dx}{x-4}. \]
Step 5: Use the basic log integral.
\[ =-\frac13\log|x-1|+\frac43\log|x-4|+c. \]
Step 6: Combine the logs.
Take $\dfrac13$ common: $\dfrac13\big[4\log|x-4|-\log|x-1|\big]+c$, which is \[ \frac13\log\left|\frac{(x-4)^4}{x-1}\right|+c. \] \[ \boxed{\dfrac13\log\left|\dfrac{(x-4)^4}{x-1}\right|+c} \]