Evaluate the indefinite integral.
integral of (x^8)/(1+x^18)
To evaluate the indefinite integral of (x^8)/(1+x^18), we can use the substitution method.
Let's substitute u = x^9. Then, du = 9x^8 dx. Solving for dx, we have dx = (1/9x^8)du.
Substituting these into the integral, we get:
= ∫ [(x^8)/(1+x^18)] dx
= ∫ [(1/9x^8)(x^8)/(1+(x^9)^2)] du
= (1/9) ∫ [1/(1+u^2)] du
Now, we can use the inverse tangent integral formula to evaluate the integral of 1/(1+u^2):
= (1/9) arctan(u) + C
Substituting back u = x^9, we have:
= (1/9) arctan(x^9) + C
Therefore, the indefinite integral of (x^8)/(1+x^18) is (1/9) arctan(x^9) + C, where C is the constant of integration.