Standard content

“Integration in Depth: Reduction Formulas and Limits of Sums” is available to Standard subscribers.

Beyond the textbook. Build reduction formulas that bring a hard integral down to an easy one, use the symmetry properties on harder integrals, and turn limits of sums into integrals. Then compare integrals without evaluating them, and differentiate an integral whose limits depend on $x$. It is at the level of JEE Advanced, A-Level Further Maths and IB Higher Level, with a full timed paper.

3Lessons12Problems2Python Labs1Question Set1Mock Test

Read the first lesson freeReduction formulas

What's inside

Reduction formulas: A reduction formula writes an integral that depends on a whole number $n$ in terms of the same kind of integral with a smaller $n$ . It is found by integration by parts or by an identity. Repeating it brings a hard integral down to an easy one. For $\int x^n e^x\,dx$

The chapter itself stays free

This is an extra course for Integrals. Its main courses are open to everyone:

Create an account

Symmetry tricks and limits of sums: Reflections turn hard integrals into easy ones: add an integral to its own reflected copy and the integrand often collapses. Harder cases use

\int_0^\pi x f(\sin x)\,dx

\int_{-a}^{a}\dfrac{f(x)}{1 + e^x}\,dx

and logarithms of trigonometric functions. A sum of the form

\dfrac{1}{n}\sum f\left(\dfrac{r}{n}\right)

becomes the integral

\int_0^1 f(x)\,dx

n \to \infty

, and this gives exact values for many limits.

Inequalities between integrals and integrals with variable limits: If

f \le g

[a, b]

, then

\int_a^b f \le \int_a^b g

. This lets us trap an integral between two numbers without finding an antiderivative. When the upper or lower limit of an integral depends on

x

, Leibniz's rule gives the derivative:

\dfrac{d}{dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))\,h'(x) - f(g(x))\,g'(x)

. It also helps with limits that have the form

\dfrac{0}{0}

, and with equations that have an integral in them.