Function by Parts in TypeScript

Introduction

In mathematics, integration by substitution or commonly known as function by parts, is a technique to find integrals by which a function is transformed into another easier function. In TypeScript, we can use this method to simplify complex integrals in our algorithms.

Syntax

The syntax of the function by parts in TypeScript, is as follows:

function integrate(f: (x: number) => number, g: (x: number) => number, a: number, b: number, n: number): number {
    let h = (b - a) / n;
    let sum = 0;
    for (let i = 1; i <= n; i++) {
        sum += ((f(a + (i - 1) * h) * g(a + i * h)) - (f(a + i * h) * g(a + (i - 1) * h))) * (h / 2);
    }
    return sum;
}

The function takes in five parameters, f, g, a, b, and n which are a function, a function, a number, a number, and a number respectively. The function returns a number.

Example Usage

Let's say we have two functions f and g like this:

function f(x: number): number {
    return 3 * x * x;
}

function g(x: number): number {
    return Math.sin(x);
}

We can then use the integrate function to find the value of the integral of product of f and g from 0 to pi:

console.log(integrate(f, g, 0, Math.PI, 1000));

This will output the value of the integral which is approximately 4.494.

Conclusion

The function by parts is a useful technique to simplify complex integrals in our algorithms. In TypeScript, we can implement this technique in our programs to solve various mathematical problems.