Question Statement

Evaluate the following indefinite integral:

$\int \left(5e^{5x}-x^{-3}+3^{2x}\right)dx$

---

Background and Explanation

To solve this problem, we use the linearity property of integrals, which allows us to integrate each term of the sum separately. We also apply standard integration rules for exponential functions ($e^{ax}$ and $a^{kx}$) and the power rule for $x^n$.

---

Solution

To find the integral, we first split the expression into three separate integrals:

$\int 5e^{5x}dx-\int x^{-3}dx+\int 3^{2x}dx$

Now, we evaluate each term individually:

Step 1: Integrate $5e^{5x}$

For the first term, we use the rule $\int e^{ax}dx=\frac{1}{a}{e}^{ax}+C$. The constant $5$ remains outside the integration process: $\int 5e^{5x}dx=5\cdot \left(\frac{1}{5}{e}^{5x}\right)=e^{5x}$

Step 2: Integrate $x^{-3}$

For the second term, we use the power rule $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$, where $n=-3$: $\int x^{-3}dx=\frac{x^{-3+1}}{-3+1}=\frac{x^{-2}}{-2}=-\frac{1}{2}{x}^{-2}$ Since there is a subtraction sign before this term in the original expression, it becomes: $-\left(-\frac{1}{2}{x}^{-2}\right)=+\frac{1}{2}{x}^{-2}$

Step 3: Integrate $3^{2x}$

For the third term, we use the general exponential rule $\int a^{kx}dx=\frac{a^{kx}}{k\ln (a)}+C$. Here, $a=3$ and $k=2$: $\int 3^{2x}dx=\frac{3^{2x}}{2\ln (3)}$

Step 4: Combine the results

Finally, we combine all the evaluated parts and add the constant of integration $C$:

$\int \left(5e^{5x}-x^{-3}+3^{2x}\right)dx=e^{5x}+\frac{1}{2}{x}^{-2}+\frac{3^{2x}}{2\ln (3)}+C$

This can also be written as: $e^{5x}+\frac{1}{2x^2}+\frac{3^{2x}}{2\ln (3)}+C$

---

Key Formulas or Methods Used

• Linearity of Integrals: $\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$
• Exponential Rule (base $e$): $\int e^{ax}dx=\frac{1}{a}{e}^{ax}+C$
• Power Rule: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ (for $n\ne -1$)
• General Exponential Rule: $\int a^{kx}dx=\frac{a^{kx}}{k\ln (a)}+C$

---

Summary of Steps

1. Split the integral into three separate parts using the linearity property.
2. Apply the exponential integration rule to $5e^{5x}$ to get $e^{5x}$.
3. Apply the power rule to $x^{-3}$ to get $-\frac{1}{2}{x}^{-2}$.
4. Apply the general exponential rule to $3^{2x}$ to get $\frac{3^{2x}}{2\ln (3)}$.
5. Combine all terms and add the integration constant $C$.