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Question

If I=(1x1/2x1/4+log(1+x1/4)x1/2+x1/4)dx, then I equals

A
2x+4x1/4+log|x1/41|+C
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B
2(x1/26x1/4+3)log(x1/4+1)+C(x1/210x1/4+7)+2(log(x1/4+1))2
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C
xx1/4+6log(x1/4+1)3x3/4+C
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D
none of these
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Solution

The correct option is D none of these
I=(1x1/2x1/4+log(1+x1/4)x1/2+x1/4)dx
Put x=t4
I=(4t3dtt2t+log(1+t)t2+14t3)dt
Let I=I1+I2
Where
I1=4t2t1dt=4(t+1+1t1)dt=2t2+4t+log|t1|=2x+4x3/4+logx1/41
And
I2=4log(1+t)t+1t2dtt+1=uI2=4loguu(u1)2du=4(u2+1u)logudu=(2u28u)logu(u28u)+2(logu)2=2(x1/26x1/4+3)log(x1/4+1)(x1/210x1/4+7)+2(log(x1/4+1))2

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