find the indefinite integral.\n∫x⁸eˣ⁹ dx\n∫x⁸eˣ⁹ dx = □

Explanation:

Step1: Use substitution

Let $u = x^{9}$, then $du=9x^{8}dx$, and $x^{8}dx=\frac{1}{9}du$.

Step2: Rewrite the integral

$\int x^{8}e^{x^{9}}dx=\int e^{u}\cdot\frac{1}{9}du$.

Step3: Integrate with respect to u

$\int e^{u}\cdot\frac{1}{9}du=\frac{1}{9}\int e^{u}du$. Since $\int e^{u}du = e^{u}+C$, we have $\frac{1}{9}e^{u}+C$.

Step4: Substitute back u

Substitute $u = x^{9}$ back, we get $\frac{1}{9}e^{x^{9}}+C$.

Answer:

$\frac{1}{9}e^{x^{9}}+C$