maths.free › Real Analysis › Improper integrals

Improper integrals

Integrals to infinity or across a singularity, defined as limits.

∫₁^∞ 1/x² dx is a limit of ∫₁^b as b → ∞, and it converges to 1; the same integral of 1/x diverges. The boundary between the two, p = 1, is the same boundary as for the p-series. The integral test is exactly this correspondence.

Przykład pracownika · integrate 1/x^2 dx from 1 to oo

Calculus · Definite-Integrals

Integrate x^(-2) from 1 to oo

\int_{1}^{\infty} \frac{1}{x^{2}}\, dx

Krok po kroku

\int_{1}^{\infty} \frac{1}{x^{2}}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
F(\infty) - F(1) = \left(0\right) - \left(-1\right)
Fundamental theorem of calculus: plug in the limits.
= 1
Simplify.

Odkryj odpowiedź

Teraz ty Wybierz problem, albo wpisz lub narysuj swój własny. Każdy krok, obraz, odpowiedź ukryta, dopóki nie zapytasz.

Zachowaj swoją pracę.

Darmowe konto dodaje notatki na każdej lekcji, zapis tego, co zakończyłeś, rozwiązane problemy w jednym miejscu, a także korepetytor, którego możesz zapytać o tę stronę. Matematyka sama jest otwarta dla wszystkich, zapisana lub nie.

Podpisz Zalogowanie

Symbole używane tutaj

Dotknij dowolny symbol pełnej definicji, obrazu i co oznacza każda litera.

Jak to zrobić?: Improper integrals

First find an antiderivative F, then evaluate F(b) − F(a).
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).
Fundamental theorem of calculus: plug in the limits.
Simplify.

Pytania, które ludzie zadają

What is the ε, δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.

Więcej w Real Analysis

Sequences and their limits Convergence of series Taylor approximation and error Functions of a complex variable Continuity and differentiability, rigorously Sequences of functions and uniform convergence