Question Statement

Evaluate the following indefinite integral:

$\int \left(x^3+\frac{1}{2x}-\frac{1}{x^3}\right)dx$

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Background and Explanation

To solve this problem, we use the linearity property of integrals, which allows us to integrate each term of the expression separately. We will apply the Power Rule for integration for terms with exponents and the specific logarithmic rule for the term involving $1/x$.

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Solution

To integrate the expression, we first rewrite the terms to make it easier to apply the integration rules. Specifically, we convert fractions into power form where applicable.

Step 1: Rewrite the integrand

We can rewrite the expression as: $\int x^{3}dx+\int \frac{1}{2x}dx-\int x^{-3}dx$

Note that $\frac{1}{2x}$ is treated as $\frac{1}{2}\cdot \frac{1}{x}$ because the power rule $x^n$ does not apply when $n=-1$.

Step 2: Integrate each term

Now, we apply the integration rules to each part:

1. For $x^3$: Using the power rule $\int x^{n}dx=\frac{x^{n+1}}{n+1}$: $\int x^{3}dx=\frac{x^{3+1}}{3+1}=\frac{x^4}{4}$

2. For $\frac{1}{2x}$: We pull out the constant $\frac{1}{2}$ and use the rule $\int \frac{1}{x}dx=\ln |x|$: $\int \frac{1}{2x}dx=\frac{1}{2}\int \frac{1}{x}dx=\frac{1}{2}\ln |x|$

3. For $-x^{-3}$: Using the power rule again: $-\int x^{-3}dx=-\left(\frac{x^{-3+1}}{-3+1}\right)=-\left(\frac{x^{-2}}{-2}\right)=\frac{1}{2x^2}$

Step 3: Combine the results

Combining all the integrated parts and adding the constant of integration $C$, we get:

$\frac{x^4}{4}+\frac{1}{2}\ln |x|+\frac{1}{2x^2}+C$

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Key Formulas or Methods Used

• Power Rule for Integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ (for $n\ne -1$)
• Logarithmic Integration Rule: $\int \frac{1}{x}dx=\ln |x|+C$
• Linearity of Integrals: $\int (f(x)\pm g(x))dx=\int f(x)dx\pm \int g(x)dx$
• Constant Multiple Rule: $\int k\cdot f(x)dx=k\int f(x)dx$

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Summary of Steps

1. Rewrite the terms of the integrand into power form (e.g., $1/x^3$ becomes $x^{-3}$).
2. Separate the integral into three individual integrals using the sum/difference rule.
3. Apply the Power Rule to the first and third terms.
4. Apply the natural log rule to the term containing $1/x$.
5. Simplify the resulting expression and add the integration constant $C$.