Question Statement

Evaluate the definite integral: ${\int }_2^4\left(e^{\frac{x}{2}}-e^{\frac{x}{4}}\right)dx$

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Background and Explanation

This problem involves integrating exponential functions where the exponent contains a linear function of $x$ (i.e., $ax$ form). You will apply the standard integration formula for $e^{ax}$ and evaluate the result using the Fundamental Theorem of Calculus.

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Solution

We solve this by applying the linearity property of integrals and the exponential integration formula.

Step 1: Use linearity to split the integral into two separate integrals: ${\int }_2^4\left(e^{\frac{x}{2}}-e^{\frac{x}{4}}\right)dx={\int }_2^4{e}^{x/2}dx-{\int }_2^4{e}^{x/4}dx$

Step 2: Apply the integration formula $\int e^{ax}dx=\frac{e^{ax}}{a}$ to each term.

For the first integral, $a=\frac{1}{2}$, so the antiderivative is $\frac{e^{x/2}}{1/2}=2e^{x/2}$. For the second integral, $a=\frac{1}{4}$, so the antiderivative is $\frac{e^{x/4}}{1/4}=4e^{x/4}$.

Using evaluation notation ${F(x)|}_a^b$ to denote $F(b)-F(a)$: $={\frac{e^{x/2}}{1/2}|}_2^4-{\frac{e^{x/4}}{1/4}|}_2^4\left(\text{Formula: }\int e^{ax}dx=\frac{e^{ax}}{a}\right)$

Simplify the constant coefficients: $=2{e^{x/2}|}_2^4-4{e^{x/4}|}_2^4$

Step 3: Evaluate at the upper limit ($x=4$) and lower limit ($x=2$): $=2\left(e^{4/2}-e^{2/2}\right)-4\left(e^{4/4}-e^{2/4}\right)$

Simplify the exponents: $=2\left(e^2-e\right)-4\left(e-e^{1/2}\right)$

Step 4: Distribute the coefficients and combine like terms: $=2e^2-2e-4e+4e^{1/2}$

$=2e^2-6e+4e^{1/2}$

Therefore, the value of the integral is $2e^2-6e+4\sqrt{e}$.

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Key Formulas or Methods Used

• Linearity of Integration: $\int (f(x)-g(x))dx=\int f(x)dx-\int g(x)dx$
• Exponential Integration: $\int e^{ax}dx=\frac{e^{ax}}{a}+C$ (for $a\ne 0$)
• Fundamental Theorem of Calculus: ${\int }_a^{b}f(x)dx={F(x)|}_a^b=F(b)-F(a)$
• Evaluation Notation: ${F(x)|}_a^b$ represents the antiderivative evaluated from $a$ to $b$

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Summary of Steps

1. Separate the integral into two parts using the difference rule for integration
2. Integrate each exponential using the formula $\int e^{ax}dx=\frac{e^{ax}}{a}$, giving coefficients $2$ and $4$ respectively
3. Apply the limits by substituting $x=4$ and $x=2$ into each antiderivative and subtracting (upper minus lower)
4. Simplify algebraically by distributing the constants and combining the $e$ terms to get the final answer $2e^2-6e+4e^{1/2}$