Question
Question: The value of \(\int _ { 0 } ^ { \sin ^ { 2 } x } \sin ^ { - 1 } \sqrt { t } d t + \int _ { 0 } ^ { ...
The value of ∫0sin2xsin−1tdt+∫0cos2xcos−1tdt is
A
2π
B
1
C
4π
D
None of these
Answer
4π
Explanation
Solution
We have
I=∫0sin2xsin−1tdt+∫0cos2xcos−1tdt
Putting t=sin2uin the first integral and t=cos2v in the second integral, we have
I=∫0xusin2udu−∫π/2xvsin2vdv
=∫0π/2usin2udu+∫π/2xusin2udu−∫π/2xvsin2vdv
I=∫0π/2usin2udu=(2−ucos2u)0π/2+21∫0π/2cos2udu
=(2−ucos2u)0π/2+41(sin2u)0π/2=4π .