# Univariate integration

## The Riemann integral

### Riemann sums

Given a function $$f(x)$$ and an interval $$[a,b]$$, we can divide $$[a,b]$$ into $$n$$ sections and calculate:

$$\sum_{j=0}^{n(b-a)}f(a+\dfrac{j}{n})$$

This is the Riemann sum.

### Riemann integral

We take the limit of the Riemann sum as $$n\rightarrow \infty$$

$$\int_a^b f(x)dx:= \lim_{n\rightarrow \infty } \sum_{j=0}^{n(b-a)} f(a+ \dfrac{j}{n} )$$

### Linearity

$$\int_a^bf(x)+g(x)dx=\lim_{n\rightarrow \infty }\sum_{j=0}^{n(b-a)}f(a+\dfrac{j}{n})+g(a+\dfrac{j}{n})$$

$$\int_a^bf(x)+g(x)dx=\lim_{n\rightarrow \infty }\sum_{j=0}^{n(b-a)}f(a+\dfrac{j}{n})+\lim_{n\rightarrow \infty }\sum_{j=0}^{n(b-a)}g(a+\dfrac{j}{n})$$

$$\int_a^bf(x)+g(x)dx=\int_a^bf(x)dx +\int_a^bg(x)dx$$

### Continuation

$$\int_a^bf(x)dx+\int_b^cf(x)dx=\lim_{n\rightarrow \infty }\sum_{j=0}^{n(b-a)}f(a+\dfrac{j}{n})+\lim_{n\rightarrow \infty }\sum_{j=0}^{n(c-b)}f(b+\dfrac{j}{n})$$

$$\int_a^bf(x)dx+\int_b^cf(x)dx=\lim_{n\rightarrow \infty }[\sum_{j=0}^{n(b-a)}f(a+\dfrac{j}{n})+\sum_{j=0}^{n(c-b)}f(b+\dfrac{j}{n})]$$

$$\int_a^bf(x)dx+\int_b^cf(x)dx=\lim_{n\rightarrow \infty }[\sum_{j=0}^{n(b-a)}f(a+\dfrac{j}{n})+\sum_{j=n(b-a)}^{n(c-b)+n(b-a)}f(b+\dfrac{j-n(b-a)}{n})]$$

$$\int_a^bf(x)dx+\int_b^cf(x)dx=\lim_{n\rightarrow \infty }[\sum_{j=0}^{n(b-a)}f(a+\dfrac{j}{n})+\sum_{j=n(b-a)}^{n(c-a)}f(a+\dfrac{j}{n})]$$

$$\int_a^bf(x)dx+\int_b^cf(x)dx=\lim_{n\rightarrow \infty }[\sum_{j=0}^{n(c-a)}f(a+\dfrac{j}{n})]$$

$$\int_a^bf(x)dx+\int_b^cf(x)dx=\int_a^cf(x)dx$$

## Definite and indefinite integrals

### Definite integrals

Definite integrals are between two points.

$$\int_0^1f(x)dx$$

### Indefinite integrals

Indefinite integrals are not. Eg +c at end. The antiderivative.

$$\int f(x)dx$$

### Unsigned definite integral

$$\int_{[0,1]}f(x)dx$$

## Anti-derivatives

### Anti-derivative

Taking the derivative of a function provides another function. The anti-derivative of a function is a function which, when differentiated, provides the original function.

As this function can include any additive constant, there are an infinite number of anti-derivatives for any function.

## Integration by parts

### Integration by parts

We have:

$$\dfrac{\delta y}{\delta x}=f(x)g(x)$$

We want that in terms of $$y$$.

We know from the product rule of differentiation:

$$y=a(x)b(x)$$

Means that:

$$\dfrac{\delta y}{\delta x}=a'(x)b(x)+a(x)b'(x)$$

So let’s relabel $$f(x)$$ as $$h'(x)$$

$$\delta$$

$$\dfrac{\delta y}{\delta x}=h'(x)g(x)$$

$$\dfrac{\delta y}{\delta x}+h(x)g'(x)=h'(x)g(x)+h(x)g'(x)$$

$$y+\int h(x)g'(x)=\int h'(x)g(x)+h(x)g'(x)$$

$$y+\int h(x)g'(x)=h(x)g(x)$$

$$y=h(x)g(x)-\int h(x)g'(x)$$

For example:

$$\dfrac{\delta y}{\delta x}=x.\cos(x)$$

$$f(x)=\cos(x)$$

$$g(x)=x$$

$$h(x)=\sin(x)$$

$$g'(x)=1$$

So:

$$y=x\int \cos(x) dx-\int \sin(x)dx$$

$$y=x\sin(x)-\cos(x)+c$$

## The fundamental theorem of calculus

### Mean value theorem for integration

Take function $$f(x)$$. From the extreme value theorem we know that:

$$\exists m \in \mathbb{R} \exists M\in \mathbb{R}\forall x\in [a,b](m<f(x)<M)$$

### Fundamental theorem of calculus

From continuation we know that:

$$\int_a^{x_1}f(x)dx+\int_{x_1}^{x_1+\delta x}f(x)dx=\int_a^{x_1+\delta x}f(x)dx$$

$$\int_x^{x_1+\delta x}f(x)dx=\int_a^{x_1+\delta }f(x)dx-\int_a^{x_1 }f(x)dx$$

Indefinite integrals

## Other

### Trigonometric substitution

For later? Haven’t defined trigonometry yet.

### Getting functions from derivatives

$$f(c)=f(a)+\int^c_a \dfrac{\delta }{\delta x}f(x) dx$$