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Question

Integrate the following functions w.r.t. x.

1x1/2+x1/3dx.


Solution

Let I=1x1/2+x1/3dxPut x1/6=t x=t6dx=6t5dt  I=6t5(t6)1/2+(t6)1/3dt=6t5t3+t2dt=6t5t2(t+1)dt=6t3(t+1)dt=6(t3+11t+1)dt=6[(t+1)(t2t+1)t+11t+1]dt=6{t2t+1+1t+1}dt=6{t33t22+tlog |t+1|}+C=6{(x1/6)33(x1/6)22+(x1/6)log |x1/6+1|}+C   ( t=x1/6)=2x1/23x1/3+6x1/66 log |x1/6+1|+C


Mathematics

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