# Antiderivative *Antidifferentiation* is a reverse process of [[Derivative|Differentiation]]. It produces an *antiderivative*. Note that because many different antiderivatives can exist for a particular function (because of the antiderivative's constant term), we cannot say that differentiation is invertible. Note that even though the antiderivative is also referred to as the *indefinite integral*, and that it uses similar notation (the elongated S), and that the antiderivative is used to compute the [[Integral|integral]], the two are very different concepts and should NOT be lumped together in my opinion because this adds unnecessary complexity to calculus newcomers. Whereas the antiderivative represents a function, the integral represents the area under a section of a curve. For the same reason, I believe it's a mistake to call antidifferentiation *integration*, but unfortunately that term is already standardized as well. ^sm5a2j ## Antiderivative Notation $f(x) \xrightarrow{\text{differentiation}}f'(x) \xrightarrow{\text{antidifferentiation}} F(x) = f(x)+C$ The antiderivative of $f'(x)$ is denoted as $F(x)$ $F(x) = \int f'(x) \, dx$ ## Antiderivative Rules Not many actual "rules" exist for antiderivatives. They closely follow basic derivative rules. ^15i5e4 $\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$ $\int Cf(x) dx = C\int f(x)dx \; \text{(where $C$ is a constant)}$ $ \int x^n dx = \begin{cases} n \ne -1 & \frac 1 {n+1}x^{n+1} + C \\ n = -1 & \ln |x| + C \end{cases} $ ^suvvl2 ## Basic Techniques for Determining Antiderivatives Where the process of differentiation is relatively simple, reversing it can be complicated. All techniques boil down to using [[Integration by Parts]] and [[Integration by Substitution]] to reverse the derivative product rule and chain rule respectively. The two can be considered basic techniques. More advanced techniques are built on those, and involve simplifying the integrant into parts that are then easier to deal with using the basic techniques. These basic techniques can be used to expand on the rules above to include: ![[Integration by Parts#^aftqxa]] ([[Integration by Parts]], reversed product rule) ![[Exponential Functions#^8n8b22]] ![[Logarithm Rules#^scpwa7]] $\int \frac {f'(x)} {f(x)} dx = \ln|f(x)| + C \; \text {(subsitute $u=f(x)$)}$ ^idur7o ## Advanced Techniques ### Trig Substitutions Integrants that contain expressions such as $a^2-x^2$ and $a^2 + x^2$ (where $a$ is a constant) can be handled by using [[Trigonometric Functions]] and specifically the [[Trigonometric Functions#Pythagorean Identity|Pythagorean Identity]]. It works by applying trigonometric substitutions: ![[Integration by Substitution#^54k49h]] ### Partial Fraction Integration ![[Rational Functions#^8s7sar]] ## Common Antiderivatives of Trigonometric Functions ![[Trigonometric Functions#^12sj15]] ## Antiderivatives of Reciprocal Quadratics ![[Antiderivatives of Reciprocal Quadratics#^vkb1ui]] ![[Antiderivatives of Reciprocal Quadratics#^w2nryz]] ![[Antiderivatives of Reciprocal Quadratics#^u28w30]] For proofs see [[Antiderivatives of Reciprocal Quadratics]].