Integration of Root Functions – Radical Integration Formulas

Integrals involving roots (especially square roots) often require specific algebraic manipulation or substitution to simplify. These standard forms help evaluate complex radical expressions efficiently.

Common Root-Based Integrals

\[ \int \frac{dx}{\sqrt{ax + b}} = \frac{2}{a} \sqrt{ax + b} + C \]
\[ \int \sqrt{ax + b} \, dx = \frac{2}{3a} (ax + b)^{\frac{3}{2}} + C \]
\[ \int \frac{x \, dx}{\sqrt{ax + b}} = \frac{2(ax - 2b)}{3a^2} \sqrt{ax + b} + C \]
\[ \int x \sqrt{ax + b} \, dx = \frac{2(3ax - 2b)}{15a^2} (ax + b)^{\frac{5}{2}} + C \]
\[ \int \frac{dx}{(x + c) \sqrt{ax + b}} = \frac{1}{\sqrt{b - ac}} \ln \left| \frac{\sqrt{ax + b} - \sqrt{b - ac}}{\sqrt{ax + b} + \sqrt{b - ac}} \right| + C, \quad (b - ac > 0) \]
\[ \int \frac{dx}{(x + c) \sqrt{ax + b}} = \frac{1}{\sqrt{ac - b}} \arctan \left( \frac{\sqrt{ax + b}}{\sqrt{ac - b}} \right) + C, \quad (ac - b > 0) \]
\[ \int x^2 \sqrt{a + bx} \, dx = \frac{2(8a^2 - 12abx + 15b^2)}{105b^3} \sqrt{a + bx}^{\,3} + C \]
\[ \int \frac{dx}{\sqrt{a^2 + x^2}} = \ln \left( x + \sqrt{x^2 + a^2} \right) + C \]
\[ \int \frac{dx}{\sqrt{x^2 - a^2}} = \ln \left( x + \sqrt{x^2 - a^2} \right) + C \]
\[ \int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C \]

Terminology

Radical: A root expression such as \( \sqrt{x} \) or \( (x^2 + a^2)^{1/2} \).
Root Integral: An integral involving square roots or fractional exponents.
Inverse Trigonometric Form: Integrals that lead to functions like arcsin or arctan.
Logarithmic Result: Some root-based expressions yield natural logarithms in their solution.

Applications

Solving integrals in geometry involving curves and arcs.
Common in physics for work, energy, and gravitational fields with radial symmetry.
Important in engineering for calculating stress or flow in parabolic and circular structures.
Used in evaluating definite integrals with bounds containing square root expressions.
Found in probability theory (e.g., in normal distribution density functions).

Integration - Integrals Involving Roots

Integrals Involving Roots

Formulas for Square Roots and Radical Expressions

Common Root-Based Integrals

Terminology

Applications

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