Evaluate the integral by making the given substitution. (Use C for the constant of integration. Remember to use absolute values where appropriate.)
integral x^5/x^6-5 dx, u = x6 − 5
I got the answer 1/6ln(x^6-5)+C but it was wrong. ps. I wrote the word integral because I couldn't find the symbol.
ok im sorry
I agree with
(1/6)ln(x^6-5)+C
To evaluate the integral, let's make the specified substitution:
u = x^6 - 5
First, let's find the derivative of u with respect to x:
du/dx = 6x^5
To express dx in terms of du, we can rearrange the equation:
dx = du / (6x^5)
Now substitute the expression for u and dx in terms of du into the original integral:
integral (x^5 / (x^6 - 5)) dx
= integral ((x^5) / (u)) * (du / (6(x^5)))
Notice that x^5 on the top and bottom cancels out, leaving:
= (1/6) * integral (du/u)
Next, evaluate the integral of 1/u with respect to u:
= (1/6) * ln|u| + C
Now, substitute back the expression for u:
= (1/6) * ln|x^6 - 5| + C
So, the correct answer to the integral, using the given substitution, is:
integral (x^5 / (x^6 - 5)) dx = (1/6) * ln|x^6 - 5| + C