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Question

Evaluate: 1011+x+xdx

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Solution

1011+x+xdx
u+v=x
dx=2xdu
=2uu2+u+1du
=(2u+12u2+u+112u2+u+1)du
For 2u+12u2+u+1du
u+v=u2+v+1
du=12u+1dv
=1vdv
=2v
=2v2+v+1
For =1u2+u+1du
=1(1+12)2+34du
u+v=2v+13
du=32dv
=1v2+1dv
=ln(v2+1+v)
=ln (2v+13)2+1+2v+13
So integral is
2x+x+1=ln(2x+x+1+x)+1)+c
Putting the values
=0.6964.

1213323_1319966_ans_ffe4e884b3b240f3a24329fffb2c7dcf.jpg

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