Evaluate the integral: ∫(1/(x-1) - 2/(x+1)) dx from -√2 to √2.
Understand the Problem
The question involves solving an integral that seems to include logarithmic functions. The user is likely looking for a step-by-step solution to the given integral expression.
Answer
$$ \ln\left( \frac{x-1}{(x+1)^2} \right) + C $$
Answer for screen readers
The integral evaluates to
$$ \ln\left( \frac{x-1}{(x+1)^2} \right) + C $$
Steps to Solve
- Break Down the Integral
We start with the integral
$$ \int \left( \frac{1}{x-1} - \frac{2}{x+1} \right) dx $$
- Separate the Integral
We can separate the integral into two parts:
$$ \int \left( \frac{1}{x-1} \right) dx - 2 \int \left( \frac{1}{x+1} \right) dx $$
- Integrate Each Part
Now, we integrate each part.
For the first part:
$$ \int \frac{1}{x-1} dx = \ln|x-1| + C_1 $$
For the second part:
$$ \int \frac{1}{x+1} dx = \ln|x+1| + C_2 $$
Thus,
$$ -2 \int \left( \frac{1}{x+1} \right) dx = -2(\ln|x+1| + C_2) = -2\ln|x+1| - 2C_2 $$
Combining these gives us:
$$ \ln|x-1| - 2\ln|x+1| + C $$
- Use Logarithmic Properties
We can simplify using properties of logarithms:
$$ \ln|x-1| - 2\ln|x+1| = \ln|x-1| - \ln|(x+1)^2| = \ln\left( \frac{x-1}{(x+1)^2} \right) $$
- Final Expression
Thus, the final result of the integral is:
$$ \int \left( \frac{1}{x-1} - \frac{2}{x+1} \right) dx = \ln\left( \frac{x-1}{(x+1)^2} \right) + C $$
The integral evaluates to
$$ \ln\left( \frac{x-1}{(x+1)^2} \right) + C $$
More Information
The integral primarily utilizes properties of logarithms and basic integration techniques. Such integrals are commonly encountered in calculus and help reinforce the understanding of logarithmic differentiation.
Tips
- Forgetting to include the constant of integration (C): Always remember to add the integration constant to your final result.
- Misapplying logarithmic properties: Make sure to carefully apply the properties of logarithms when simplifying.
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