Definite Integration Question 380
Question: $ \int_0^{2}{\frac{x^{3}dx}{{{(x^{2}+1)}^{\frac{3}{2}}}}}= $
Options:
A) $ {{(\sqrt{2}-1)}^{2}} $
B) $ \frac{{{(\sqrt{2}-1)}^{2}}}{\sqrt{2}} $
C) $ \frac{\sqrt{2}-1}{\sqrt{2}} $
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
Put $ t=x^{2}+1\Rightarrow dt=2xdx $
$ \int_0^{2}{\frac{x^{3}}{{{(x^{2}+1)}^{3/2}}}dx=\frac{1}{2}}\int_1^{5}{\frac{(t-1)}{{t^{3/2}}}dt=\frac{1}{2}\int_1^{5}{[{t^{-1/2}}-{t^{-3/2}}]dt}} $
$ =\frac{1}{2}[ 2\sqrt{t}+2\frac{1}{\sqrt{t}} ]_1^{5}=\frac{1}{2}[ 2\sqrt{5}+\frac{2}{\sqrt{5}}-2-2 ] $
$ =[ \sqrt{5}+\frac{1}{\sqrt{5}}-2 ]=\frac{6-2\sqrt{5}}{\sqrt{5}} $ .
BETA
