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Question

x2+cos2xx2+1 cosec2x dx is equal to

A
cotxcot1x+c
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B
ccotx+cot1x
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C
tan1x cosec xsecx+c
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D
elogtan1xcotx+c
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Solution

The correct options are
B ccotx+cot1x
C tan1x cosec xsecx+c
D elogtan1xcotx+c
Let
I=x2+cos2xx2+1cosec2 xdx
=x2+11+cos2xx2+1cosec2 xdx
=(1sin2xx2+1)cosec2 xdx
=(cosec2 x1x2+1)dx
=cotxtan1x+c(1)
=cotx+cot1xπ2+c
=cotx+cot1x+c

From (1),
I=tan1xcosec xsecx+c

Again from (1)
I=elogtan1xcotx+c

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