Definite Integration Question 189
Question: $ \int_0^{\pi /2}{\sqrt{\cos \theta }{{\sin }^{3}}\theta }d\theta = $
Options:
A) $ \frac{20}{21} $
B) $ \frac{8}{21} $
C) $ \frac{-20}{21} $
D) $ \frac{-8}{21} $
Show Answer
Answer:
Correct Answer: B
Solution:
Let $ I=\int_0^{\pi /2}{\sqrt{\cos \theta }}{{\sin }^{3}}\theta d\theta $
Put $ t=\cos \theta \Rightarrow dt=-\sin \theta d\theta , $ then I = $ -\int_1^{0}{{t^{1/2}}(1-t^{2})dt=\int_0^{1}{({t^{1/2}}-{t^{5/2}})}} $
$ dt $
I = $ [ \frac{2}{3}{t^{3/2}}-\frac{2}{7}{t^{7/2}} ]_0^{1}=\frac{8}{21} $ .
BETA
