Maths WorkshopMaths problems with full solutions

Improper integrals

12 problems · Integrals on unbounded intervals or with unbounded integrands.

Ex.5When do integrals of powers converge?
#improper integrals #p-test #Riemann integrals
Improper integrals
L1
L1
Ex.7An integral whose integrand blows up
#improper integrals #logarithm #limits
Improper integrals
L1
L1
Ex.21An improper integral with exponentials
#improper integral #exponential #partial fraction
Improper integrals
L1
L1
Ex.29An improper integral of a rational function
#improper integrals #arctangent #convergence
Improper integrals
M1
M1
Ex.53Proving convergence by comparison
#improper integrals #comparison test #divergence
Improper integrals
L1
L1
Ex.67A convergent improper integral of $\ln x/(1+x)^2$
#improper integral #integration by parts #logarithm
Improper integrals
L1
L1
Ex.68An improper integral of $\ln x/x^2$ by parts
#improper integral #integration by parts #logarithm
Improper integrals
L1
L1
Ex.102When the equivalence rule for improper integrals fails
#improper integral #integration by parts #absolute convergence #conditional convergence
Improper integrals
L1
L1
Ex.321An improper integral
#improper integrals #integration by parts
Improper integrals
L2
L2
Ex.729An integral singular at both endpoints
#improper integrals #substitution #arcsine
Improper integrals
L1
L1
Ex.783Differentiating under the integral sign
#parameter integrals #Gamma function #factorial
Improper integrals
M1
M1
Ex.2077The integral of sin x / x
#improper integrals #conditional convergence #integration by parts
Improper integrals
M1
M1