Algebraic Integration Techniques

The simplest algebraic technique is expanding a product or power before integrating. The Power Rule states that ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1. This rule applies to each individual term, so expanding first is essential. For example, ∫(x² − 3)(x + 2) dx requires expanding the product: (x² − 3)(x + 2) = x³ + 2x² − 3x − 6, giving ∫(x³ + 2x² − 3x − 6) dx = x⁴/4 + 2x³/3 − 3x²/2 − 6x + C. Never try to integrate a product of polynomials term-by-term without expanding first — that gives a wrong answer.

When the integrand is a rational function where the numerator has degree greater than or equal to the denominator, divide each term of the numerator by the denominator separately. For ∫(3x³ + 4x − 1)/x² dx, split into ∫(3x + 4x⁻¹ − x⁻²) dx = 3x²/2 + 4 ln|x| + x⁻¹ + C. This works because integration is linear: ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx. For proper rational functions (numerator degree < denominator degree), use polynomial long division first to extract an integer part, then handle the remainder using partial fractions or standard logarithmic forms.

Sometimes an integrand contains a common factor in numerator and denominator that can be cancelled before integrating. For instance, ∫(x² − 4)/(x − 2) dx. Factor the numerator: x² − 4 = (x − 2)(x + 2), so the expression simplifies to x + 2 (valid when x ≠ 2). Then ∫(x + 2) dx = x²/2 + 2x + C. This is an algebraic simplification, not a substitution, and it should always be attempted when you see a factorable numerator. Note: cancellation is valid for all x in the domain where the original expression is defined.

When a radical appears in the integrand, algebraic manipulation can remove it or simplify it. For expressions like ∫(√x + 1)/√x dx, rewrite as ∫(1 + x⁻¹/²) dx = x + 2√x + C by splitting and using the power rule with n = −1/2. For integrals like ∫1/(1 + √x) dx, multiply numerator and denominator by the conjugate (1 − √x) to clear the radical: (1 − √x)/((1 + √x)(1 − √x)) = (1 − √x)/(1 − x), then split into simpler terms. Rewriting nth roots as fractional exponents (∛x = x^(1/3)) is often the quickest rationalization approach.

Completing the square transforms a quadratic expression ax² + bx + c into the form a(x − h)² + k, which is essential for integrals that lead to inverse-trigonometric or logarithmic antiderivatives. For ∫1/(x² + 6x + 13) dx, complete the square: x² + 6x + 13 = (x + 3)² + 4. The integral becomes ∫1/((x+3)² + 4) dx. Using the standard formula ∫1/(u² + a²) du = (1/a) arctan(u/a) + C with u = x + 3 and a = 2 gives (1/2) arctan((x+3)/2) + C. Without completing the square, this integral would appear unsolvable to most students.

• f(x) — First numerator function (dimensionless): One term in the numerator of the rational integrand
• g(x) — Second numerator function (dimensionless): Another term in the numerator of the rational integrand
• h(x) — Denominator function (dimensionless): The common denominator shared by all split fractions; must be nonzero on the interval of integration
• dx — Differential (units of x): Indicates the variable of integration and corresponds to an infinitesimally small width in a Riemann sum

Problem. Evaluate ∫(x³ + 5x² − 2x + 4) / x² dx

Answer. x²/2 + 5x − 2 ln|x| − 4/x + C

• Power Rule for Integration — Algebraic simplification techniques are used specifically to reduce integrands to power-rule form, so understanding ∫xⁿ dx = xⁿ⁺¹/(n+1) + C is the direct payoff of these algebraic manipulations.
• Integration by Substitution (u-substitution) — Completing the square and rationalizing are often preliminary steps before applying u-substitution, making these algebraic techniques a natural bridge to more advanced integration methods.
• Partial Fraction Decomposition — After splitting a rational function and performing polynomial long division, remaining proper rational functions are handled by partial fractions — the next level of algebraic integration technique.
• Polynomial Long Division — Long division of polynomials is the core algebraic tool used to convert improper rational functions into a polynomial plus a proper remainder, a prerequisite step for integrating many rational expressions.
• Completing the Square — Completing the square is the algebraic technique that unlocks integrals leading to arctan and natural logarithm antiderivatives, directly connecting algebraic manipulation to inverse-trigonometric integration forms.