Question

\( \int_0^1 \frac{1}{(x^2+16)(x^2+25)} \, dx = \)

• A. \( \frac{1}{5}\tan^{-1}\frac{1}{4} - \frac{1}{4}\tan^{-1}\frac{1}{5} \)
• B. \( \frac{1}{9}\left[\frac{1}{4}\tan^{-1}\frac{1}{4} - \frac{1}{5}\tan^{-1}\frac{1}{5}\right] \)
• C. \( \frac{1}{4}\tan^{-1}\frac{1}{4} - \frac{1}{5}\tan^{-1}\frac{1}{5} \)
• D. \( \frac{1}{9}\left[\frac{1}{5}\tan^{-1}\frac{1}{4} - \frac{1}{4}\tan^{-1}\frac{1}{5}\right] \)
• E. \( \frac{1}{9}\left[\frac{3}{4}\tan^{-1}\frac{1}{4} - \frac{4}{5}\tan^{-1}\frac{1}{5}\right] \)

Hint:

For expressions like \( (x^2+a^2)(x^2+b^2) \), always try partial fractions immediately.

Answer 1

The Correct Option is B