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Question

Integrate with respect to x:
(i) x31x3+x (ii) 1x2+2x+5

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Solution

(i) (x31)dxx3+x

=[(x31)(x+1)x3+x]dx

=[1x+1x(x2+1)]dx

=dx(x+1)x(x+1)dx

=dxxdxx(x2+1)dxx(x2+1)

By using 1a2+x2dx=1atan1(xa)+c

=xtan1(x)[(x2+1)x2]x(x2+1)dx

=xtan1xdxx+xx2+1dx

=xtan1xlnx+12ln(x2+1)+C

(x31)(x3+x)=xtan1xln(x2+1x)+C


(ii) dxx2+2x+5

=dx(x+1)2+(2)2

=ln(x+(x+1)2+(2)2)+C

=ln(x+x2+2x+5)+C

dxx2+2x+5=ln(x+x2+2x+5)+C

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