Definite Integration Question 231
Question: $ \int_0^{\infty }{\frac{x^{3}dx}{{{(x^{2}+4)}^{2}}}=} $
Options:
A) 0
B) $ \infty $
C) $ \frac{1}{2} $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ \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 }{[ \frac{1}{t+4}-\frac{4}{{{(t+4)}^{2}}} ]dt=\frac{1}{2}[ \log (t+4)+\frac{4}{t+4} ]_0^{\infty }} $
$ =\frac{1}{2}[ \log \infty +0-(\log 4+1) ]=\infty $ .
BETA
