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Problem 11

# Evaluate the integrals. $$\int x^{-5} d x$$

Expert verified
The short answer for the given integral is: $$\int x^{-5} dx = -\frac{1}{4}x^{-4} + C$$.
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## Step 1: Recognize the format of the integrand

Here, we have an integrand of the form $$x^n$$ with $$n = -5$$. The power rule for integration can be applied to this type of function.

## Step 2: Apply the power rule for integration

The power rule for integration states that: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ Where $$C$$ is the constant of integration. In our case, $$n = -5$$, so we apply the rule as follows: $\int x^{-5} dx = \frac{x^{-5+1}}{-5+1} + C = \frac{x^{-4}}{-4} + C$

## Step 3: Write the final answer

Our final answer for this integral is: $\int x^{-5} dx = -\frac{1}{4}x^{-4} + C$

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