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Question

(x+1)2x2+3 dx is equal to
(where C is integration constant)

A
13(x2+3)32+x22x2+3+322lnx+x2+23+C
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B
16(2x2+3)32+x22x2+3+322lnx+x2+32+C
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C
16(x23)34+x2x23+322lnx+x2+32+C
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D
13(x26)34+x2x26+322lnx+x2+32+C
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Solution

The correct option is B 16(2x2+3)32+x22x2+3+322lnx+x2+32+C
I=(x+1)2x2+3dx
Let x+1=λddx(2x2+3)+μ
x+1=4xλ+μ
Equating coefficient of same degree term, we get
λ=14,μ=1I=(14(4x)+1)2x2+3dxI=144x2x2+3dx+2x2+3dx

Putting 2x2+3=t4x dx=dt
Now,
I=14t dt+2x2+3dx =t326+ 2[x2+(32)2]dxI=16(2x2+3)32+2x2+(32)2dxI=16(2x2+3)32+x22x2+3+322lnx+x2+32+C

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