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Given
I=1/√x(1+x^1/3)…(1)
Put 1+x^1/3 =t
Differentiate w.r.t.x
0+1/3 x^-2/3 =dt/dx
Or 1/3x^2/3 =dt/dx
Or dx/√x =3xdt
From (1)
I=3xdt/t
=3(t-1)^3dt/t
=3[(t^3 -3t^2 +3t -1)dt/t]
=3[(t^2 -3t+3–1/t)dt]
integrate
I=3[t^3 /3 -3t^2 /2 +3t-ln|t|]+c
Where t=(1+x^1/3)
❤️
I=1/√x(1+x^1/3)…(1)
Put 1+x^1/3 =t
Differentiate w.r.t.x
0+1/3 x^-2/3 =dt/dx
Or 1/3x^2/3 =dt/dx
Or dx/√x =3xdt
From (1)
I=3xdt/t
=3(t-1)^3dt/t
=3[(t^3 -3t^2 +3t -1)dt/t]
=3[(t^2 -3t+3–1/t)dt]
integrate
I=3[t^3 /3 -3t^2 /2 +3t-ln|t|]+c
Where t=(1+x^1/3)
❤️
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