Elementary Antiderivatives

Antiderivatives

Definition: Antiderivative

An antiderivative of the real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is any differentiable function $F:D\subseteq \mathbb{R}\to \mathbb{R}$ whose derivative is $f$.

$$F^{\prime }(x)=f(x)$$

Theorem: Family of Antiderivatives

Let $F$ be an antiderivative of $f$.

Any other differentiable function $\mathcal{F}:D\subseteq \mathbb{R}\to \mathbb{R}$ is also an antiderivative of $f$ if and only if there is some constant $C\in \mathbb{R}$ such that

$\mathcal{F}(x)=F(x)+C\forall x\in D$

PROOF

TODO

Indefinite Integrals

Definition: Indefinite Integral

The indefinite integral of a real function $f$ is the set of all antiderivatives of $f$.

$$\int f(x)\mathrm{dx}\stackrel{\text{def}}{=}\{F\mid F^{\prime }=f\}$$

NOTATION

The notation $\int f(x)\mathrm{dx}$ is often used to denote a particular antiderivative of $f$ or simply the process of integration, as well.

Since antiderivatives are unique up to a constant, if we know a particular antiderivative $F$ of $f$, we write

$$\int f(x)\mathrm{dx}=F(x)+C$$

for the indefinite integral.

THEOREM

If $f$ and $g$ are continuous on a closed interval $[a;b]$, then for all $\alpha ,\beta \in \mathbb{R}$

$\int \alpha f(x)+\beta g(x)\mathrm{dx}=\alpha \int f(x)\mathrm{dx}+\beta \int g(x)\mathrm{dx}$

PROOF

Let $F(x)$ be an antiderivative $f(x)$ and $G(x)$ be an antiderivative of $g(x)$.

Let’s examine the derivative of the right-hand side.

$$\frac{\mathrm{d}}{\mathrm{d}x}\left(\lambda \int f(x)\mathrm{dx}+\mu \int g(x)\mathrm{dx}\right)$$

Since differentiation is linear, we have

$$\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}x}\left(\lambda \int f(x)\mathrm{dx}+\mu \int g(x)\mathrm{dx}\right)& =\lambda \frac{\mathrm{d}}{\mathrm{d}x}\int f(x)\mathrm{dx}+\mu \frac{\mathrm{d}}{\mathrm{d}x}\int g(x)\mathrm{dx}\\ & =\lambda \frac{\mathrm{d}}{\mathrm{d}x}F(x)+\mu \frac{\mathrm{d}}{\mathrm{d}x}G(x)\\ & =\lambda f(x)+\mu g(x)\end{array}$$

This means that $\lambda F(x)+\mu G(x)$ is an antiderivative of $\lambda f(x)+\mu g(x)$. All other antiderivatives of $\lambda f(x)+\mu g(x)$ can be expressed as $\lambda F(x)+\mu G(x)+C$ for some $C\in \mathbb{R}$.

Theorem: Integration by Parts

If $u$ and $v$ are continuously differentiable, then

$\int u(x)v^{\prime }(x)\mathrm{dx}=u(x)v(x)+\int u^{\prime }(x)v(x)\mathrm{dx}$

PROOF

TODO

Theorem: Integration by Substitution

$$\int f(\underset{u}{\underbrace{g(x)}})\underset{\mathrm{du}}{\underbrace{g^{\prime }(x)\mathrm{dx}}}=\int f(u)\mathrm{du}$$

PROOF

TODO

THEOREM

$$\int \frac{f^{\prime }(x)}{f(x)}\mathrm{dx}=\ln (f(x))+C,f(x)>0$$

PROOF

TODO

Theorem: Antiderivatives of Polynomial Functions

$$\int x^{\alpha }\mathrm{dx}=\frac{1}{\alpha +1}{x}^{\alpha +1}+C\forall \alpha \ne -1\in \mathbb{R}$$ $$\int (ax+b{)}^{\gamma }\mathrm{dx}=\frac{(ax+b{)}^{\gamma +1}}{a(\gamma +1)}+C$$

PROOF

TODO

Theorem: Antiderivatives of Rational Functions

$\int \frac{c}{ax+b}\mathrm{dx}=\frac{c}{a}\ln |ax+b|+C$

For all $n\in \mathbb{N}$ and $a\in \mathbb{R}$:

$\int \frac{\mathrm{dx}}{(x-a{)}^n}=\{\begin{array}{l}\ln |x-a|+C,\text{ if }n=1\\ -\frac{1}{(n-1)(x-a{)}^{n-1}}+C,\text{ if }n\ge 2\end{array}$

For all $a,b\in \mathbb{R}$ and all polynomials $x^2+px+q$ with $p^2-4q<0$:

$\int \frac{\mathrm{dx}}{x^2+px+q}=\frac{2}{\sqrt{4q-p^2}}\arctan \frac{2x+p}{\sqrt{4q-p^2}}+C$

$\int \frac{ax+b}{x^2+px+q}\mathrm{dx}=\frac{a}{2}\ln (x^2+px+q)-\left(b-\frac{ap}{2}\right)\int \frac{\mathrm{dx}}{x^2+px+q}$

For all $n\ge 2\in \mathbb{N}$ and all polynomials $x^2+px+q$ with $p^2-4q<0$:

$\int \frac{\mathrm{dx}}{(x^2+px+q{)}^n}=\frac{2x+p}{(n-1)(4q-p^2)(x^2+px+q{)}^{n-1}}+\frac{2(2n-3)}{(n-1)(4q-p^2)}\int \frac{\mathrm{dx}}{(x^2+px+q{)}^{n-1}}$

$\int \frac{ax+b}{(x^2+px+q{)}^n}\mathrm{dx}=-\frac{a}{2(n-1)(x^2+px+q{)}^{n-1}}+\left(b-\frac{ap}{2}\right)\int \frac{\mathrm{dx}}{(x^2+px+q{)}^{n-1}}$

PROOF

TODO

Theorem: Antiderivatives of Exponential Functions

$\int e^{\alpha x}\mathrm{dx}=\frac{1}{\alpha }{e}^{\alpha x}+C$

$\int f^{\prime }(x)e^{f(x)}\mathrm{dx}=e^{f(x)}+C$

$\int a^x\mathrm{dx}=\frac{a^x}{\ln a}+C$

$\int e^x(f(x)+f^{\prime }(x))\mathrm{dx}=e^{x}f(x)+C$

PROOF

TODO

Theorem: Antiderivatives of Logarithmic Functions

$\int \ln x\mathrm{dx}=x(\ln x-1)+C$

$\int {\log }_{a}x\mathrm{dx}=\frac{x}{\ln a}(\ln x-1)+C$

PROOF

TODO

Theorem: Antiderivatives of Trigonometric Functions

$$\int \sin x\mathrm{dx}=-\cos x+C$$ $$\int \cos x\mathrm{dx}=\sin x+C$$ $$\int \tan x\mathrm{dx}=-\ln |\cos x|+C$$ $$\int \cot x\mathrm{dx}=\ln |\sin x|+C$$ $$\int \arctan x\mathrm{dx}=x\arctan x-\frac{1}{2}\ln (x^2+1)+C$$

PROOF

TODO