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Question

tan1xdx=__________+c.

A
xtan1x+12logtan1xx2+1
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B
xtan1x12logx2+1
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C
xtan1x+12logx2+1
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D
11+x2
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Solution

The correct option is A xtan1x12logx2+1
I=tan1x dx
Here taking u=tan1x and v=1
dudx=11+x2 and vdx=1dx=x
I=uvdx(uvdx)dx
(Rule of integration by parts)
=(tan1x)xx1+x2dx
=x(tan1x)122x1+x2dx
I=xtan1x12f(x)f(x)dx,
Where f(x)=1+x2
=xtan1x12log|f(x)|+c
I=xtan1x12log|1+x2|+c.

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