List of effective methods for solving definite and improper integrals:
Efficient methods for solving definite integrals:
- Substitution method:
- Если ( u = g(x) ), то ( \int f(g(x)) g'(x) \, dx = \int f(u) \, du ).
- Integration by parts method:
- Formula: ( \int u \, dv = uv – \int v \, du ).
- Variable replacement:
- Using trigonometric, exponential, or other substitutions to simplify the integral.
- Symmetry:
- Using the symmetry properties of a function to simplify calculations.
- Numerical methods:
- Trapezoid, Simpson and other methods for approximate calculation of integrals.
Efficient methods for solving improper integrals:
- 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.
- Comparison test:
- Comparison with known integrals to determine convergence or divergence.
- Variable replacement:
- Using suitable substitutions to simplify the integral, as in the case of definite integrals.
- Decomposition into simple fractions:
- For rational functions, to simplify the integral.
- Special features:
- Using functions such as the Gamma function or the Beta function to solve integrals that cannot be expressed as elementary functions.