Find the integral.  `int (5x^4)/(7+x^5)^4 dx`

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txmedteach | High School Teacher | (Level 3) Associate Educator

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To evaluate this integral, we will most easily start by making a substitution. Let's let `u = 7+x^5`. Now, we can find what the differential of `u` will be by taking the derivative with respect to `x`:

`(du)/(dx) = 5x^4`

Multiply both sides by the differential of `x`, `dx`, to get the following result:

`du = 5x^4dx`

Notice that this is the entire numerator of the integral! We can therefore substitute the part inside the parentheses with `u` and we can substitute `5x^4dx` with `du`:

`int(5x^4)/(7+x^5)^4 dx = int (du)/u^4`

Let's restate the integral on the right to get us a more familiar form:

`int(du)/u^4 = int u^-4 du`

Now, we can use the following common integral relation to solve:

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

Recall, `C` is the constant of integration. Let's apply this relation to our situation:

`int u^-4 du = u^-3/-3 + C`


`= -1/(3u^3) + C`

Recall, `u = 7 + x^5`. We can now substitute back into the equation this function of `x` to get our final result:

` =` `-1/(3(7+x^5)^3) + C`

In summary, we found that:

`int(5x^4)/(7+x^5)^4 dx = -1/(3(7+x^5)^3) + C`

I hope this helps!