Question

$\int_{0}^{\infty}{\frac{x^{3}dx}{(x^{2} + 4)^{2}} =}$

• A. 0
• B. $\infty$
• C. $\frac{1}{2}$
• D. None of these

Answer 1

Answer: (B)

$\infty$

Answer 2

$\int_{0}^{\infty}{\frac{x^{3}dx}{(x^{2} + 4)^{2}} = \frac{1}{2}}\int_{0}^{\infty}{\frac{x^{2}2xdx}{(x^{2} + 4)^{2}}dx} = \frac{1}{2}\int_{0}^{\infty}{\frac{t}{(t + 4)^{2}}dt}$,

[Putting$x^{2} = t$]

$= \frac{1}{2}\int_{0}^{\infty}{\left\lbrack \frac{1}{t + 4} - \frac{4}{(t + 4)^{2}} \right\rbrack dt = \frac{1}{2}\left\lbrack \log(t + 4) + \frac{4}{t + 4} \right\rbrack_{0}^{\infty}}$

$= \frac{1}{2}\left\lbrack \log\infty + 0 - (\log 4 + 1) \right\rbrack = \infty$.