Improper integral

Limits of integration:

from to

Find the integral!

v

The graph:

from to

Piecewise:

{ enter the piecewise function here

What can the calculator of improper integrals do?

Analysis of the improper integral:

• Detect of convergence
• Divergence analysis
• Consideration of conditional convergence and divergence
• Consideration of absolute convergence and divergence

Calculating of improper integral:

• Calculation of a first-type integral with infinite lower or upper limits, as well as with two infinite limits
• Calculation of the integral of the 2nd kind from functions with uncertainty or divergence at certain points
• Consideration of conditionally and absolutely convergent integrals

Using the calculator with the application of methods:

• Application of the comparison test I
• Using the comparison test II
• Application of Dirichlet's principle
• Working with the Newton-Leibniz formula for the improper integral of the second kind (antiderivative)

Function analysis considering:

• Monotonicity on the integration interval (increasing or decreasing)
• Sign of functions (positive or negative values on the interval)
• The presence of discontinuities (approaching infinity) or continuity
• The alternation of functions

Examples of improper integrals

The above examples also contain:

• the modulus or absolute value: absolute(x) or |x|
• square roots sqrt(x),
  cubic roots cbrt(x)
• trigonometric functions:
  sinus sin(x), cosine cos(x), tangent tan(x), cotangent ctan(x)
• exponential functions and exponents exp(x)
• inverse trigonometric functions:
  arcsine asin(x), arccosine acos(x), arctangent atan(x), arccotangent acot(x)
• natural logarithms ln(x),
  decimal logarithms log(x)
• hyperbolic functions:
  hyperbolic sine sh(x), hyperbolic cosine ch(x), hyperbolic tangent and cotangent tanh(x), ctanh(x)
• inverse hyperbolic functions:
  hyperbolic arcsine asinh(x), hyperbolic arccosinus acosh(x), hyperbolic arctangent atanh(x), hyperbolic arccotangent acoth(x)
• other trigonometry and hyperbolic functions:
  secant sec(x), cosecant csc(x), arcsecant asec(x), arccosecant acsc(x), hyperbolic secant sech(x), hyperbolic cosecant csch(x), hyperbolic arcsecant asech(x), hyperbolic arccosecant acsch(x)
• rounding functions:
  round down floor(x), round up ceiling(x)
• the sign of a number:
  sign(x)
• for probability theory:
  the error function erf(x) (integral of probability), Laplace function laplace(x)
• Factorial of x:
  x! or factorial(x)
• Gamma function gamma(x)
• Lambert's function LambertW(x)
• Trigonometric integrals: Si(x), Ci(x), Shi(x), Chi(x)

The insertion rules

The following operations can be performed

2*x

- multiplication

3/x

- division

x^2

- squaring

x^3

- cubing

x^5

- raising to the power

x + 7

- addition

x - 6

- subtraction

Real numbers

insert as 7.5, no 7,5

Constants

pi

- number Pi

e

- the base of natural logarithm

i

- complex number

oo

- symbol of infinity