Questions: Find Int(e^x, x, 3,7)

Find Int(e^x, x, 3,7)
Transcript text: Find $\operatorname{Int}\left(e^{\wedge} \mathrm{x}, \mathrm{x}, 3,7\right)$
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Solution

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Solution Steps

To find the integral of a function, we need to use a numerical integration method or a symbolic computation library. In this case, we will use the sympy library in Python to compute the definite integral of the function \( e^x \) from 3 to 7.

Step 1: Define the Integral

We are tasked with finding the definite integral of the function \( e^x \) over the interval from 3 to 7. This can be expressed as:

\[ \int_{3}^{7} e^x \, dx \]

Step 2: Compute the Antiderivative

The antiderivative of \( e^x \) is itself, \( e^x \). Therefore, the indefinite integral of \( e^x \) is:

\[ \int e^x \, dx = e^x + C \]

where \( C \) is the constant of integration.

Step 3: Evaluate the Definite Integral

To find the definite integral from 3 to 7, we evaluate the antiderivative at the upper and lower limits and subtract:

\[ \left[ e^x \right]_{3}^{7} = e^7 - e^3 \]

Step 4: Simplify the Expression

The result of the definite integral is:

\[ e^7 - e^3 \]

This expression represents the exact value of the integral.

Final Answer

The value of the definite integral is:

\[ \boxed{e^7 - e^3} \]

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