Introduction to Indefinite Integration

In differential calculus we learned how to find the derivative of a function. Indefinite integration asks the reverse question: given a function $f(x)$, can we find a function $g(x)$ whose derivative is $f(x)$? If $g'(x)=f(x)$, we call $g(x)$ a primitive (or antiderivative) of $f(x)$, and we write

read as "the integral of $f(x)$ with respect to $x$ is $g(x)+c$". Here $f(x)$ is called the integrand and $c$ is an arbitrary constant, called the constant of integration.

The constant $c$ is essential, not optional: since the derivative of any constant is $0$, if $g(x)$ is one primitive of $f(x)$ then $g(x)+c$ is a primitive too, for every real number $c$. So $\int f(x)\,dx$ does not denote a single function but an entire family of curves, all differing from each other by a vertical shift, and all sharing the same tangent slope at any given $x$-value.

For example, $\frac{d}{dx}(x^3)=3x^2$, so $\int 3x^2\,dx = x^3+c$. The curves $y=x^2$, $y=x^2+4$ and $y=x^2-5$ are three members of the family $y=x^2+c$. At $x=2$ their $y$-values are $4$, $8$ and $-1$ respectively — different points — but the slope of the tangent at each of those points is the same, $2(2)=4$, because they all share the derivative $2x$.

This is the geometric picture behind indefinite integration: finding $\int f(x)\,dx$ means finding the whole family of curves whose tangent-slope function is $f(x)$.

The systematic theory of integration was developed independently by Sir Isaac Newton and Gottfried Leibnitz. In this chapter we restrict attention to methods for finding indefinite integrals of algebraic, trigonometric, exponential and logarithmic functions.