Questions: Evaluate the integral. [ int0^1(9 x^e+2 e^x) dx ]

Transcript text: Evaluate the integral. \[ \int_{0}^{1}\left(9 x^{e}+2 e^{x}\right) d x \]

Solution

Solution Steps

Step 1: Define the Integral

We need to evaluate the integral

\[ \int_{0}^{1}\left(9 x^{e}+2 e^{x}\right) d x. \]

Step 2: Break Down the Integral

The integral can be expressed as the sum of two separate integrals:

\[ \int_{0}^{1} 9 x^{e} \, dx + \int_{0}^{1} 2 e^{x} \, dx. \]

Step 3: Evaluate Each Integral

For the first integral, we evaluate

\[ \int_{0}^{1} 9 x^{e} \, dx = 9 \cdot \frac{x^{e+1}}{e+1} \bigg|_{0}^{1} = 9 \cdot \frac{1^{e+1}}{e+1} - 9 \cdot \frac{0^{e+1}}{e+1} = \frac{9}{e+1}. \]

For the second integral, we evaluate

\[ \int_{0}^{1} 2 e^{x} \, dx = 2 \cdot e^{x} \bigg|_{0}^{1} = 2 \cdot (e^{1} - e^{0}) = 2(e - 1). \]

Step 4: Combine the Results

Now, we combine the results of both integrals:

\[ \int_{0}^{1}\left(9 x^{e}+2 e^{x}\right) d x = \frac{9}{e+1} + 2(e - 1). \]

Final Answer

\(\boxed{\frac{9}{e+1} + 2(e - 1)}\)

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