Find the integral ∫ (2x + 1)/(x^2 + x) dx.

Find the integral ∫ (2x + 1)/(x^2 + x) dx.

Step 1: Factor the denominator. The expression x^2 + x can be factored as x(x + 1).
Step 2: Set up the partial fraction decomposition. We want to express (2x + 1)/(x^2 + x) as A/x + B/(x + 1) for some constants A and B.
Step 3: Multiply both sides by the denominator x(x + 1) to eliminate the fraction: 2x + 1 = A(x + 1) + Bx.
Step 4: Expand the right side: 2x + 1 = Ax + A + Bx.
Step 5: Combine like terms: 2x + 1 = (A + B)x + A.
Step 6: Set up a system of equations by comparing coefficients: A + B = 2 and A = 1.
Step 7: Solve the system of equations. From A = 1, substitute into A + B = 2 to find B = 1.
Step 8: Rewrite the integral using the values of A and B: ∫ (2x + 1)/(x^2 + x) dx = ∫ (1/x + 1/(x + 1)) dx.
Step 9: Integrate each term separately: ∫ (1/x) dx + ∫ (1/(x + 1)) dx.
Step 10: The integrals are ln|x| and ln|x + 1| respectively, so we have ln|x| + ln|x + 1| + C.
Step 11: Combine the logarithms using the property ln(a) + ln(b) = ln(ab): ln|x(x + 1)| + C.
Step 12: Recognize that x(x + 1) = x^2 + x, so the final answer is ln|x^2 + x| + C.

Partial Fraction Decomposition – This technique is used to break down a rational function into simpler fractions that can be integrated more easily.
Integration of Rational Functions – Understanding how to integrate functions that are the ratio of polynomials.
Natural Logarithm in Integration – Recognizing that the integral of a function can result in a logarithmic expression.