How can the function dy/dx=x^y be integrated?

Given the function:

`(dy)/(dx) = x^y`

It can be integrated as:

`int(dy) = int x^y dx`

We can use the power rule of integration to solve this case. The power rule is

`int x^n = (x^(n+1))/(n+1)`

where, n is a non-zero number.

Assuming that y `!=` 0, the given integral can be solved as:

`int x^y dx = x^(y+1) / (y+1) + C`

Thus, the power rule helps us in getting the solution to the given integral.

We can also calculate the value of C (the constant of integration), if we are provided with some limits, or some values of y, given values of x.

For example, let us say when x =1, y =1.

In that case, y = 1 = x^(y+1) / (y+1) + C = 1^2 / (1 + 1) + C

or, C = 1/2.

Hope this helps.