Riemann sums

The fundamental theorem of calculus

The idea behind the fundamental theorem is that a definite integral can be computed using an antiderivative.

For a function f(x), the antiderivative (aka indefinite integral) is commonly denoted F(x) and is defined to be the value that, when differentiated, gives f(x).

\frac{d}{dx}F(x) = f(x)
F(x) = \int f(x)
\int_{a}^b f(x) = F(b)-F(a)

Single integrals

An integral is an infinite sum. $$ \int_{a}^b x^2 dx $$

Computing integrals

Integration by substitution

Integration by parts

The product rule for differentiation states that

\frac{d}{dx}(h(x)g(x)) = \frac{dh}{dx}g + h\frac{dg}{dx}

integrating this equation on both sides with respect to x yields

\int \frac{d}{dx}(h(x)g(x)) = \int \frac{dh}{dx}g dx + \int h\frac{dg}{dx} dx

as the indefinite integral is defined to be the inverse operation to differentiation, the left hand side of the equation simplifies to h(g)g(x). On the right hand side the differentials cancel, leaving

h(x)g(x) = \int g dh + \int h dg

simple algebra shows that

\int g dh = h(x)g(x) - \int h dg

this formula is known as integration by parts.

Integration using partial fractions

More interesting approaches

Differentiating under the integral sign
Bring it over method

Multiple integrals