Questions: Evaluate the following integral. [ int-2^2 e^9 x+7 d x ]

Evaluate the following integral. [ int-2^2 e^9 x+7 d x ]
Transcript text: Evaluate the following integral. \[ \int_{-2}^{2} e^{9 x+7} d x \]
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Solution

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Solution Steps

Step 1: Set Up the Integral

We start with the integral we need to evaluate: \[ \int_{-2}^{2} e^{9x + 7} \, dx \]

Step 2: Evaluate the Integral

Using the properties of exponents, we can rewrite the integrand: \[ e^{9x + 7} = e^7 \cdot e^{9x} \] Thus, the integral becomes: \[ e^7 \int_{-2}^{2} e^{9x} \, dx \] The integral of \(e^{9x}\) is: \[ \frac{1}{9} e^{9x} \] Evaluating this from \(-2\) to \(2\): \[ \left[ \frac{1}{9} e^{9x} \right]_{-2}^{2} = \frac{1}{9} e^{18} - \frac{1}{9} e^{-18} \]

Step 3: Combine Results

Now, we can combine the results: \[ e^7 \left( \frac{1}{9} e^{18} - \frac{1}{9} e^{-18} \right) = \frac{1}{9} e^7 \left( e^{18} - e^{-18} \right) \] This simplifies to: \[ \frac{1}{9} e^7 \left( e^{18} - e^{-18} \right) = \frac{1}{9} \left( e^{25} - e^{-11} \right) \]

Final Answer

Thus, the value of the integral is: \[ \boxed{\frac{1}{9} \left( e^{25} - e^{-11} \right)} \]

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