A) \[\frac{1}{2}\pi x+c\]
B) \[x({{\sin }^{-1}}x-{{\cos }^{-1}}x)+c\]
C) \[x({{\cos }^{-1}}x+{{\sin }^{-1}}x)+c\]
D) \[\frac{\pi }{2}+x+c\]
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{({{\sin }^{-1}}x+{{\cos }^{-1}}x)\,dx}=\int_{{}}^{{}}{\left( \frac{\pi }{2} \right)}\,dx=\frac{\pi x}{2}+c\] \[\left( \because \,\,\,{{\sin }^{-1}}x+{{\cos }^{-1}}x=\frac{\pi }{2} \right)\] Also \[\int_{{}}^{{}}{({{\sin }^{-1}}x+{{\cos }^{-1}}x)dx=x({{\cos }^{-1}}x+{{\sin }^{-1}}x)+c}\].
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