Explore the Wonders of Calculus: A Guide to ∫ 1/(1 – x²) dx

Step 1: Recognize the Integral

We want to find the integral of the expression 1/(1 – x²): ∫ 1/(1 – x²) dx

Step 2: Factor the Denominator

Factor the denominator as a difference of squares: 1 – x² = (1 + x)(1 – x)

Step 3: Apply Partial Fraction Decomposition

Express the integrand as a sum of partial fractions: A/(1 + x) + B/(1 – x)

Step 4: Find the Values of A and B

Use algebraic methods to find the constants A and B that satisfy the equation.

Step 5: Integrate the Partial Fractions

Integrate each partial fraction separately: ∫ A/(1 + x) dx + ∫ B/(1 – x) dx

Step 6: Write the Final Result

Combine the results of the integrals to write the final expression for the antiderivative of 1/(1 – x²).

Conclusion: The integral of 1/(1 – x²) is a classic problem in calculus that requires understanding of factoring, partial fraction decomposition, and integration techniques. Mastering this integral is a rewarding mathematical achievement!