We have to find the integral of f(x)=x^7/(x^8+4) = y

Let x^8 + 4 = t

=> dt/dx = 8*x^7

=> dt/ 8 = x^7 dx

Noe y = x^7/(x^8+4)

Int [ x^7/(x^8+4) dx]

=> Int [(1/8)* (1/t) dt]

=> (1/8) ln t + C

substitute t = x^8 + 4

=> (1/8) ln(x^8 + 4) + C

**Therefore the integral of x^7/(x^8+4) is (1/8) ln(x^8 + 4) + C**

Evaluate the indefinite integral of f(x)=x^7/(x^8+4).

We put x^8+4 = t.

We differentiate x^8+4 = t.

=> (x^8+4)' = 8x^7 dx = dt.

=> x^7 dx = dt/8.

We substitute x^7 dx = dt/8 in Int f(x) = Int x^7 dx/(x^8+4)+c

=> Int f(x) dx = Int dt/8t+C

=> int f(x) dx = (1/8) logt.+C

We replace t = x^8+4.

=>int f(x) dx = (1/8)log(x^8+4)+C.

Therefore Int f(x) dx = (1/8)log(x^8+4)+C.

We'll solve the indefinite integral using substitution method. We'll change the denominator x^8+4 = t.

We'll differentiate both sides:

8x^7dx = dt

x^7dx = dt/8

We'll re-write the integral:

Int x^7dx/(x^8+4) = Int (dt/8)/t

Int x^7dx/(x^8+4) = (1/8)*Int dt/t

Int x^7dx/(x^8+4) = (1/8)* ln |t| + C

Int x^7dx/(x^8+4) = (1/8)* ln(x^8+4) + C

**Int x^7dx/(x^8+4) = ln(x^8+4)^(1/8) + C**