Question

evaluate the indefinite integral.\\(int x^{8}e^{x^{9}}dx\\)\\(\boxed{}\\) + c

Explanation:

Step1: Choose substitution variable

Let $u = x^9$

Step2: Compute derivative of u

$\frac{du}{dx} = 9x^8 \implies du = 9x^8 dx \implies x^8 dx = \frac{1}{9}du$

Step3: Substitute into integral

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

Step4: Integrate with respect to u

$\frac{1}{9} \int e^u du = \frac{1}{9}e^u$

Step5: Substitute back u

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

Answer:

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