Question Statement

Evaluate the following indefinite integral:

$\int \left(x^{\frac{3}{2}}+e^{3x}+x^0\right)dx$

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Background and Explanation

To solve this problem, we use the linearity property of integration, which allows us to integrate each term in the expression individually. We will apply the power rule for integration and the specific rule for integrating exponential functions with a linear constant in the exponent.

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Solution

To find the integral, we break the expression into three separate parts:

$\int x^{\frac{3}{2}}dx+\int e^{3x}dx+\int x^{0}dx$

Step 1: Integrate $x^{\frac{3}{2}}$

Using the power rule $\int x^{n}dx=\frac{x^{n+1}}{n+1}$, where $n=\frac{3}{2}$: $\int x^{\frac{3}{2}}dx=\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}=\frac{x^{\frac{5}{2}}}{\frac{5}{2}}$ Simplifying the fraction: $\frac{2}{5}{x}^{\frac{5}{2}}$

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

Using the exponential rule $\int e^{ax}dx=\frac{1}{a}{e}^{ax}$, where $a=3$: $\int e^{3x}dx=\frac{1}{3}{e}^{3x}$

Step 3: Integrate $x^0$

First, recognize that any non-zero base raised to the power of zero is 1 ($x^0=1$): $\int x^{0}dx=\int 1dx$ The integral of a constant is: $x$

Step 4: Combine the results

Now, we combine all the integrated parts and add the constant of integration $C$: $\frac{2}{5}{x}^{\frac{5}{2}}+\frac{1}{3}{e}^{3x}+x+C$

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Key Formulas or Methods Used

• Power Rule for Integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$
• Exponential Rule: $\int e^{ax}dx=\frac{1}{a}{e}^{ax}+C$
• Linearity Property: $\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$
• Zero Exponent Rule: $x^0=1$

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Summary of Steps

1. Split the integral into three individual terms.
2. Apply the power rule to $x^{\frac{3}{2}}$ by adding 1 to the exponent and dividing by the new exponent.
3. Apply the exponential rule to $e^{3x}$ by dividing by the coefficient of $x$.
4. Simplify $x^0$ to 1 and integrate it to get $x$.
5. Sum the results and include the integration constant $C$.