Question

$\int \frac{2 x^2-1}{x^4-x^2-20} d x=$

• A. $\frac{1}{5} \log \left|\frac{x+5}{x-5}\right| + \tan^{-1}\left(\frac{x}{2}\right) + c$
• B. $\frac{1}{2\sqrt{5}} \log \left|\frac{x+\sqrt{5}}{x-\sqrt{5}}\right| + \tan^{-1}\left(\frac{x}{2}\right) + c$
• C. $\frac{1}{2\sqrt{5}} \log \left|\frac{x-\sqrt{5}}{x+\sqrt{5}}\right| + \frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right) + c$
• D. $\frac{1}{2} \log \left|\frac{x-\sqrt{5}}{x+\sqrt{5}}\right| + \frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right) + c$

Hint:

When every $x$ term in an integral is raised to an even power, substituting $x^2=t$ algebraically (without changing $dx$) makes finding the partial fractions incredibly simple. Just don't forget to swap $x^2$ back in before integrating!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to evaluate an indefinite integral with an even polynomial in both the numerator and the denominator.

Step 2: Key Formula or Approach:
Use partial fractions by temporarily substituting $x^2 = t$ to split the integrand. Then, integrate the resulting terms using standard formulas: $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left|\frac{x-a}{x+a}\right|$ and $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right)$.

Step 3: Detailed Explanation:
Let the integrand be $f(x) = \frac{2x^2 - 1}{x^4 - x^2 - 20}$.
Temporarily substitute $x^2 = t$ just for algebraic manipulation:
$\frac{2t - 1}{t^2 - t - 20} = \frac{2t - 1}{(t - 5)(t + 4)}$
Apply partial fractions:
$\frac{2t - 1}{(t - 5)(t + 4)} = \frac{A}{t - 5} + \frac{B}{t + 4}$
$2t - 1 = A(t + 4) + B(t - 5)$
Put $t = 5$: $2(5) - 1 = A(9) \implies 9 = 9A \implies A = 1$.
Put $t = -4$: $2(-4) - 1 = B(-9) \implies -9 = -9B \implies B = 1$.
So, the expression splits into: $\frac{1}{t - 5} + \frac{1}{t + 4}$.
Replace $t$ back with $x^2$ and integrate:
$I = \int \left( \frac{1}{x^2 - 5} + \frac{1}{x^2 + 4} \right) dx$
$I = \int \frac{1}{x^2 - (\sqrt{5})^2} dx + \int \frac{1}{x^2 + 2^2} dx$
Using the standard integral formulas:
$I = \frac{1}{2\sqrt{5}} \log \left|\frac{x - \sqrt{5}}{x + \sqrt{5}}\right| + \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + c$

Step 4: Final Answer:
The result matches option (C).