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Question

Integrate x(x2+3)+3(tan1x)(1+x2)2(1+x2)dx (where C is an integration constant)

A
(x2+3)(1+x2)tan1x+C
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B
x(x2+3)tan1x+C
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C
tan1x1+x2+C
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D
none
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Solution

The correct option is C x(x2+3)tan1x+C
I=x(x2+3)+3(tan1x)(1+x2)21+x2dx
=[x(x2+3)1+x2+3(tan1x)(1+x2)]dx
=[x+2x1+x2+3(tan1x)+3x2tan1x]dx
=x22+ln(1+x2)+3(1+x2)tan1xdx
=x22+ln(1+x2)+3[(x+x33)tan1xx(3+x2)3(1+x2)dx]+c
=x22+ln(1+x2)+(3x+x3)tan1xx22ln(1+x2)+c
=(x3+3x)tan1x+c
=x(x2+3)tan1x+c
Hence, the answer is x(x2+3)tan1x+c.


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