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Useful integrals and linearity

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Contrary to the previous section, where we built intution by using approximate integration, in this section we will focus on computing integrals analytically. We will start by looking at the integrals of simple functions and then we will see how linearity helps us compute integrals of slightly more complex functions.

Some simple integrals

The following is a list of useful integrals. As you go through the integrals below, take a moment to make sure the results make sense from a differentiation perspective, i.e. if you differentiate the result, do you get the function we are trying to integrate?

Constant function:∫a dx=ax+CPower function:∫xa dx=xa+1a+1+C, ∀x≠−1∈RExponential function:∫ax dx=axln⁡a+CSine function:∫sin⁡x dx=−cos⁡x+CCosine function:∫cos⁡x dx=sin⁡x+C\begin{align} \textrm{Constant function}:& \int a \,\textrm{d}x = ax + C\\[3 ex] \textrm{Power function}:& \int x^a \,\textrm{d}x = \frac{x^{a+1}}{a+1} + C, \,\forall x \neq -1 \in \mathbb{R} \\[3 ex] \textrm{Exponential function}:& \int a^x \,\textrm{d}x = \frac{a^x}{\ln a} + C\\[3 ex] \textrm{Sine function}:& \int \sin x \,\textrm{d}x = -\cos x + C\\[3 ex] \textrm{Cosine function}:& \int \cos x \,\textrm{d}x = \sin x + C\\[3 ex] \end{align}

To compute the integral of more complicated functions, we need to learn the basic rules of integration.

Linearity in integration

The first really useful rule is that, akin to differentiation, integration is linear. That means that integration respects the two following rules:

Addition/Subtraction:∫[f(x)±g(x)] dx=∫f(x) dx±∫g(x) dxScalar multiplication:∫cf(x) dx=c∫f(x) dx\begin{align} \textrm{Addition/Subtraction}:& \int [f(x) \pm g(x)] \,\textrm{d}x = \int f(x) \,\textrm{d}x \pm \int g(x) \,\textrm{d}x\\[3 ex] \textrm{Scalar multiplication}:& \int cf(x) \,\textrm{d}x = c\int f(x) \,\textrm{d}x \end{align}

Thus, the integral of a linear combination of functions is the same as the linear combination of the integrals of those functions, and we can write

∫[af(x)±bg(x)] dx=a∫f(x) dx±b∫g(x) dx.\begin{equation} \int [af(x) \pm bg(x)] \,\textrm{d}x = a\int f(x) \,\textrm{d}x \pm b\int g(x) \,\textrm{d}x . \end{equation}

Again, these rules are also observed for differentiation. In the next sections we will look at two other rules: integration by parts and change of variables, which are the integration counterpart of the product rule and the chain rule in differentiation.