Integral Calculus Question 237
Question: If $ \phi (x)=\int{{{\cot }^{4}}xdx+\frac{1}{3}{{\cot }^{3}}x-\cot x} $ and $ \phi ( \frac{\pi }{2} )=\frac{\pi }{2} $ then $ \phi (x) $ is
Options:
A) $ \pi -x $
B) $ x-\pi $
C) $ \pi /2-x $
D) x
Show Answer
Answer:
Correct Answer: D
Solution:
[d] $ \int{{{\cot }^{4}}xdx=\int{{{\cot }^{2}}x.( \cos ec^{2}x-1 )dx}} $ $ =\int{{{\cot }^{2}}x\cos ec^{2}xdx-\int{(\cos ec^{2}x-1)dx}} $ $ =-\frac{1}{3}{{\cot }^{3}}x+\cot x+x+c $
$ \therefore \phi (x)=-\frac{1}{3}{{\cot }^{3}}x+\cot x+x+c+\frac{1}{3} $ $ {{\cot }^{3}}x-\cot x $ $ =x+c $
$ \therefore \phi ( \frac{\pi }{2} )=\frac{\pi }{2}+c,\therefore c=0,\therefore \phi (x)=x $
BETA
