Question Statement

Evaluate the indefinite integral:

$\int \left(e^{\frac{9}{2}x}+\frac{1}{x}\right)dx$

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Background and Explanation

This problem requires integrating the sum of an exponential function with a linear coefficient and the reciprocal function. You will need to apply the linearity property of integration to handle each term separately, using standard integration formulas for exponential and logarithmic functions.

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Solution

Let $I$ denote the given integral:

$I=\int \left(e^{\frac{9}{2}x}+\frac{1}{x}\right)dx$

Using the linearity property of integrals (the integral of a sum equals the sum of the integrals), we can split this into two separate integrals:

$I=\int e^{\frac{9}{2}x}dx+\int \frac{1}{x}dx$

Step 1: Evaluate the exponential integral

For the term $\int e^{\frac{9}{2}x}dx$, we use the rule for integrating exponential functions of the form $e^{ax}$. The general formula states that $\int e^{ax}dx=\frac{e^{ax}}{a}+C$, where $a$ is the coefficient of $x$. Here, $a=\frac{9}{2}$:

$\int e^{\frac{9}{2}x}dx=\frac{e^{\frac{9}{2}x}}{\frac{9}{2}}$

Dividing by $\frac{9}{2}$ is equivalent to multiplying by $\frac{2}{9}$:

$\frac{e^{\frac{9}{2}x}}{\frac{9}{2}}=\frac{2}{9}{e}^{\frac{9}{2}x}$

Step 2: Evaluate the reciprocal integral

For the term $\int \frac{1}{x}dx$, we apply the standard logarithmic integration rule:

$\int \frac{1}{x}dx=\ln x$

(Note: This assumes $x>0$. For the general case including negative values, this would be $\ln |x|$.)

Step 3: Combine the results

Adding both integrated terms together and including the constant of integration $c$:

$I=\frac{2}{9}{e}^{\frac{9x}{2}}+\ln x+c$

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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: $\int e^{ax}dx=\frac{e^{ax}}{a}+C$ (where $a\ne 0$)
• Logarithmic integration: $\int \frac{1}{x}dx=\ln |x|+C$ (or $\ln x+C$ when $x>0$)

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Summary of Steps

1. Apply linearity to separate the integral into two parts: $\int e^{\frac{9}{2}x}dx$ and $\int \frac{1}{x}dx$
2. Integrate the exponential by dividing $e^{\frac{9}{2}x}$ by its coefficient $\frac{9}{2}$, yielding $\frac{2}{9}{e}^{\frac{9}{2}x}$
3. Integrate the reciprocal to obtain $\ln x$
4. Combine all terms and add the arbitrary constant $c$ to get the final answer: $\frac{2}{9}{e}^{\frac{9x}{2}}+\ln x+c$