Integration

60 problems · Integrals of functions of one variable.

Search problems Level

Ex.5When do integrals of powers converge?

#improper integrals #p-test #Riemann integrals

Improper integrals

Ex.7An integral whose integrand blows up

#improper integrals #logarithm #limits

Improper integrals

Ex.19Step functions and their integrals

#step function #partition #integral

Riemann integral

High school

Ex.20Integral of a constant step function

#constant #step function #elementary

Riemann integral

High school

Ex.21An improper integral with exponentials

#improper integral #exponential #partial fraction

Improper integrals

Ex.29An improper integral of a rational function

#improper integrals #arctangent #convergence

Improper integrals

Ex.45Additivity over adjacent intervals

#additivity #step function #split

Riemann integral

High school

Ex.46A step function built from the floor

#step function #floor function #integral

Riemann integral

High school

Ex.47Linearity of the integral on step functions

#linearity #step functions #sum

Riemann integral

High school

Ex.48Monotonicity of the integral

#monotonicity #step functions #comparison

Riemann integral

High school

Ex.49Summary of step functions and Riemann integration

#summary #step functions #Riemann integrability

Riemann integral

High school

Ex.50Two subdivisions for the same function

#step function #subdivision #independence

Riemann integral

High school

Ex.53Proving convergence by comparison

#improper integrals #comparison test #divergence

Improper integrals

Ex.63Integral of $|f|$ for a step function

#absolute value #step function #triangle inequality

Riemann integral

High school

Ex.64Darboux sums for a step function

#Darboux sums #upper lower #step functions

Riemann integral

Ex.65A step function with zero integral

#zero integral #step functions #nonnegative

Riemann integral

High school

Ex.66Squeezing between step functions

#Riemann integrability #step functions #squeeze

Riemann integral

Ex.67A convergent improper integral of $\ln x/(1+x)^2$

#improper integral #integration by parts #logarithm

Improper integrals

Ex.68An improper integral of $\ln x/x^2$ by parts

#improper integral #integration by parts #logarithm

Improper integrals

Ex.75Composition with a continuous function

#composition #continuous #integrability preservation

Riemann integral

Ex.76A monotone function on a fine grid

#monotone functions #Riemann integrability #step approximation

Riemann integral

Ex.96A parameter integral equal to arctan

#parameter integral #fundamental theorem #arctan

Riemann integral

Ex.97A limit via the mean value theorem for integrals

#mean value theorem for integrals #logarithm #limit

Riemann integral

Ex.102When the equivalence rule for improper integrals fails

#improper integral #integration by parts #absolute convergence #conditional convergence

Improper integrals

Ex.103Limit of an integral with arctan

#mean value theorem for integrals #arctan #limit

Riemann integral

Ex.104Limit of an integral with cosine and logarithm

#mean value theorem for integrals #cosine #logarithm

Riemann integral

Ex.117Limits of sums as Riemann sums

#Riemann sums #limits #pi #ln 2

Riemann integral

Ex.119A limit of a product via Riemann sums

#Riemann sums #logarithm #limit #product

Riemann integral

Ex.187Integral of sine squared

#integrals #trigonometry

Integration techniques

Ex.188Integration by parts

#integrals #by parts

Integration techniques

Ex.199Area between a parabola and a line

#area #definite integrals #intersection points

Riemann integral

High school

Ex.200An area with an absolute value

#definite integrals #absolute value #sign of a function

Riemann integral

High school

Ex.202Recognising antiderivatives

#antiderivatives #definite integrals #fundamental theorem

Riemann integral

High school

Ex.210Integrals that simplify by parity

#symmetry #odd and even functions

Riemann integral

High school

Ex.321An improper integral

#improper integrals #integration by parts

Improper integrals

Ex.374Integration by parts, two classics

#integration by parts #definite integrals

Integration techniques

Ex.381Simplifying and decomposing rational integrands

#partial fractions #rational functions

Integration techniques

Ex.386Integration by substitution

#substitution #change of variable

Integration techniques

Ex.394The derivative of an integral with variable bounds

#fundamental theorem of calculus #chain rule #limits

Riemann integral

Ex.399Trigonometric integrals by linearisation

#trigonometric integrals #linearisation

Integration techniques

Ex.723A reduction formula

#reduction formula #integration by parts #recurrence

Integration techniques

Ex.727Solving an integral equation

#integral equations #fundamental theorem #differential equation

Riemann integral

Ex.728Completing the square before integrating

#rational functions #completing the square #arctangent

Integration techniques

Ex.729An integral singular at both endpoints

#improper integrals #substitution #arcsine

Improper integrals

Ex.732Comparing the harmonic sum with the logarithm

#integral comparison #harmonic series #monotonicity

Riemann integral

Ex.738Integrating by parts twice and coming back

#integration by parts #exponential #trigonometric

Integration techniques

Ex.739Darboux sums of the Dirichlet function

#Riemann integrability #Darboux sums #density

Riemann integral

Ex.763Integrating ln and arctan by parts

#integration by parts #logarithm #arctangent

Integration techniques

Ex.780The area of an ellipse

#trigonometric substitution #area #ellipse

Integration techniques

Ex.781The Cauchy–Schwarz inequality for integrals

#inequalities #Cauchy-Schwarz #quadratic polynomial

Riemann integral

Ex.783Differentiating under the integral sign

#parameter integrals #Gamma function #factorial

Improper integrals

Ex.786Using the substitution x → a + b − x

#symmetry #substitution #trigonometric integrals

Integration techniques

Ex.1472Bounds for n! by comparing with an integral

#factorial #integral comparison #Stirling

Riemann integral

Ex.1492Integrability with one wild point

#Riemann integrability #Darboux criterion #oscillation

Riemann integral

Ex.1493Limits of integrals depending on n

#limits of integrals #squeeze theorem #integration by parts

Riemann integral

Ex.1517The tangent half-angle substitution

#Weierstrass substitution #trigonometric integrals #rational functions

Integration techniques

Ex.1521Wallis integrals

#Wallis integrals #recurrence #asymptotics #pi

Integration techniques

Ex.2068An absolute value integral with a parameter

#absolute value #parameters #trigonometry

Riemann integral

Olympiad

Ex.2077The integral of sin x / x

#improper integrals #conditional convergence #integration by parts

Improper integrals

Ex.2078An integral that equals its own mirror image

#symmetry #substitution #logarithm #tangent

Integration techniques

Olympiad