3.6 Q-14

Question Statement

Evaluate the following integral: $\int_{0}^{2}\left(e^{x / 2}-e^{-x / 2}\right) , dx$

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Background and Explanation

This integral involves the difference of two exponential functions, $e^{x/2}$ and $e^{-x/2}$. To solve this, we can split the integral into two separate terms and handle each term individually. The key concept here is to evaluate the integrals of exponential functions, which is straightforward using basic integration rules.

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Solution

1. Split the Integral:

   The original integral can be split into two parts: $\int_{0}^{2} \left(e^{x/2} - e^{-x/2}\right) , dx = \int_{0}^{2} e^{x/2} , dx - \int_{0}^{2} e^{-x/2} , dx$

2. Integrate the First Term:

   To integrate $e^{x/2}$, apply the following rule: $\int e^{ax} , dx = \frac{e^{ax}}{a}$

   For $e^{x/2}$, the exponent is $a = \frac{1}{2}$, so: $\int e^{x/2} , dx = 2 e^{x/2}$

   Now, evaluate the integral from 0 to 2: $2 \left[e^{x/2}\right]_0^2 = 2\left(e^{2/2} - e^{0/2}\right) = 2(e - 1)$

3. Integrate the Second Term:

   For the second term, $e^{-x/2}$, the exponent is $a = -\frac{1}{2}$, so: $\int e^{-x/2} , dx = -2 e^{-x/2}$

   Now, evaluate this integral from 0 to 2: $-2 \left[e^{-x/2}\right]_0^2 = -2\left(e^{-2/2} - e^{0/2}\right) = -2\left(\frac{1}{e} - 1\right)$

4. Combine the Results:

   Now, add the results from both integrals: $2(e - 1) + 2\left(\frac{1}{e} - 1\right)$

   Simplifying further: $2(e - 1 + \frac{1}{e} - 1) = 2\left(e + \frac{1}{e} - 2\right)$

5. Final Expression:

   The final simplified result is: $\frac{2}{e}(e - 1)^2$

   So, the value of the integral is: $\boxed{\frac{2}{e}(e - 1)^2}$

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Key Formulas or Methods Used

• Integration of Exponential Functions: $\int e^{ax} , dx = \frac{e^{ax}}{a}$

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Summary of Steps

1. Split the integral into two parts: $\int_{0}^{2} e^{x/2} , dx \quad \text{and} \quad \int_{0}^{2} e^{-x/2} , dx$

2. Integrate each term separately:

   • For $e^{x/2}$, integrate to get $2(e - 1)$.
   • For $e^{-x/2}$, integrate to get $-2\left(\frac{1}{e} - 1\right)$.

3. Combine the results to get: $2(e - 1 + \frac{1}{e} - 1) = 2(e + \frac{1}{e} - 2)$

4. Simplify the result to get: $\frac{2}{e}(e - 1)^2$