Question

evaluate the integral \\( \int 27e^{9x + 8} dx \\).
\\( \int 27e^{9x + 8} dx = \square \\)

Explanation:

Step1: Use Constant Multiple Rule

The integral of a constant multiple of a function is the constant multiple of the integral, so we can factor out the 27: $\int 27e^{9x + 8}dx = 27\int e^{9x + 8}dx$.

Step2: Use Substitution

Let $u = 9x + 8$, then $du = 9dx$, so $dx=\frac{1}{9}du$. Substitute into the integral: $27\int e^{u}\cdot\frac{1}{9}du$.

Step3: Simplify and Integrate

Simplify the constants: $27\cdot\frac{1}{9}=3$, so we have $3\int e^{u}du$. The integral of $e^u$ is $e^u + C$, so $3e^u + C$.

Step4: Substitute Back

Replace $u$ with $9x + 8$: $3e^{9x + 8}+C$.

Answer:

$3e^{9x + 8}+C$