Integral Calculus Question 83
Question: If $ \int x^{5}{{(1+x^{3})}^{2/3}}dx=A{{(1+x^{3})}^{8/3}}+B{{(1+x^{3})}^{5/3}}+c $ ,then
Options:
A) $ A=\frac{1}{4},B=\frac{1}{5} $
B) $ A=\frac{1}{8},B=-\frac{1}{5} $
C) $ A=-\frac{1}{8},B=\frac{1}{5} $
D) none of these
Show Answer
Answer:
Correct Answer: B
Solution:
[b] Here, $ \int{x^{5}{{(1+x^{3})}^{2/3}}dx} $ Let $ 1+x^{3}=t^{2} $ and $ 3x^{2}dx=2tdt $
$ \therefore \int{x^{5}{{(1+x^{3})}^{2/3}}dx=\int{x^{3}{{(1+x^{3})}^{2/3}}x^{2}dx}} $ $ =\int{(t^{2}-1){{(t^{2})}^{2/3}}x^{2}dx} $ $ =\frac{2}{3}\int{(t^{2}-1){t^{7/3}}dt} $ $ =\frac{2}{3}\int{({t^{13/3}}-{t^{7/3}}),dt} $ $ =\frac{2}{3}{ \frac{3}{16}{t^{16/3}}-\frac{3}{10}{t^{10/3}} }+C $ $ =\frac{1}{8}{{(1+x^{3})}^{8/3}}-\frac{1}{5}{{(1+x^{3})}^{5/3}}+C $
BETA
