List of mathematical formulas: “Efficient methods for solving definite and improper integrals

List of effective methods for solving definite and improper integrals:

Efficient methods for solving definite integrals:

1. Substitution method:

• Если ( u = g(x) ), то ( \int f(g(x)) g'(x) \, dx = \int f(u) \, du ).

1. Integration by parts method:

• Formula: ( \int u \, dv = uv – \int v \, du ).

1. Variable replacement:

• Using trigonometric, exponential, or other substitutions to simplify the integral.

1. Symmetry:

• Using the symmetry properties of a function to simplify calculations.

1. Numerical methods:

• Trapezoid, Simpson and other methods for approximate calculation of integrals.

Efficient methods for solving improper integrals:

1. Limits of integration:

• Application of limits: ( \int_a^bf(x) \, dx = \lim_{c \to b} \int_a^cf(x) \, dx ) for integrals with infinite limits.

1. Comparison test:

• Comparison with known integrals to determine convergence or divergence.

1. Variable replacement:

• Using suitable substitutions to simplify the integral, as in the case of definite integrals.

1. Decomposition into simple fractions:

• For rational functions, to simplify the integral.

1. Special features:

• Using functions such as the Gamma function or the Beta function to solve integrals that cannot be expressed as elementary functions.