Given y= x^3/ (x^4 +10^5

We need to find the integral of y.

==> intg y = intg ( x^3/(x^4 +1)^5

Let us assume that u = x^4 + 1 ==> du = 4x^3 dx

Now we will substitute.

==> intg y = intg ( x^3 / u^5) * du/ 4x^3

We will reduce similar terms.

==> intg y = intg ( du/ 4u^5 )

= (1/4) intg u^-5 du

= (1/4) u^-4 / -4 + C

= (-1/16) u^-4 + C

= -1/16u^4 + C

Now we will substitute with u = x^4 +1

**==> intg y = (-1/16)*(x^4+1)^4 + C**

We'll solve the integral using substitution technique. We'll note x^4 + 1 = t.

We'll differentiate both sides:

4x^3dx = dt

We'll put:

x^3dx = dt/4

We'll write the integral in t:

Int dt/4t^5 = (1/4)Int t^-5dt

(1/4)Int t^-5dt = (1/4)*t^(-5+1)/(-5+1) +C

Int dt/4t^5 = -(1/16*t^4) + C

We'll substitute t = x^4 + 1

**Int x^3*dx/(x^4+1)^5 = -1/16*(x^4 + 1)^4 + C**