9)Evaluate the indefinite integral.
∫x^8e^(x^9)dx
let u = x^9
then du = 9x^8 dx
and the integrand is now
1/9 e^u du
To evaluate the indefinite integral ∫x^8e^(x^9)dx, we can use a substitution method.
Let u = x^9, then we can find du by differentiating both sides with respect to x:
du = 9x^8 dx
Rewriting the integral using this substitution, we have:
∫x^8e^(x^9)dx = (1/9) ∫e^udu
Now, the integral has been simplified to ∫e^udu, which is a basic integral with respect to u. The antiderivative of e^u is simply e^u, so we can evaluate this integral as follows:
(1/9) ∫e^udu = (1/9)e^u + C
Substituting back the value of u, we get:
(1/9)e^(x^9) + C
Therefore, the indefinite integral of ∫x^8e^(x^9)dx is (1/9)e^(x^9) + C, where C is the constant of integration.