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.

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