# Integration¶

### 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.