Infi-2 1

Integrals

\begin{matrix} Let f be a function on [a, b] or any other \\ Function F is called primitive of f if \\ F^{'} = f \\ For example: \\ f (x) = x^{2} ⟹ \forall C \in R : F (x) = \frac{x^{3}}{3} + C \end{matrix}

\begin{matrix} Let f be a function \\ Let F, G be antiderivatives of f \\ Then \exists C \in R : F = G + C \\ Proof: \\ (F - G)^{'} = F^{'} - G^{'} = f - f = 0 \\ ⟹ F - G is constant ⟹ \exists C \in R : F - G = C \\ ⟹ \exists C \in R : F = G + C \end{matrix}

\begin{matrix} Let f be a function \\ Integral is a set of primitives of f \\ Integral is denoted as \int f (x) d x \\ Note: this integral is an indefinite integral \end{matrix}

\begin{matrix} Let f be a function \\ We want to find it’s primitive F \end{matrix}

\begin{matrix} Let there be four types of functions \\ 1. & Functions without a primitive \\ By Darboux’s theorem, if function has a removable or a jump discontinuity \\ it has no primitive \\ 2. & Functions with "immediate" (known) integrals \\ \begin{matrix} \int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C \\ \int x^{- 1} d x = \ln | x | + C \\ \int a^{x} d x = \frac{a^{x}}{\ln a} + C \\ \int \sin (x) d x = - \cos (x) + C \\ \int \cos (x) d x = \sin (x) + C \\ \int \frac{1}{\cos^{2} (x)} d x = \tan (x) + C \\ \int \frac{1}{x^{2} + 1} d x = \arctan (x) + C \\ \int \frac{1}{\sqrt{1 - x^{2}}} d x = \arcsin (x) + C \\ \int \ln (x) d x = x \ln x - x + C \end{matrix} \\ 3. & Functions with no elementary primitive \\ For example: \int e^{x^{2}} d x = \frac{\sqrt{π}}{2} e r f i (x) + C \\ 4. & ... \end{matrix}

\begin{matrix} \int (a f + g) (x) d x = a \int f (x) d x + \int g (x) d x \\ Proof: \\ Let F^{'} = f \\ Let G^{'} = g \\ ⟹ (a F^{'} + G^{'}) = a f + g \\ ⟹ \int (a f + g) (x) d x = (a F + G) = a F + G = a \int f (x) d x + \int g (x) d x \\ ⟹ \int (a f + g) (x) d x = a \int f (x) d x + \int g (x) d x \end{matrix}

\begin{matrix} Let f, g be functions \\ Then \int (f g^{'}) (x) d x = f g - \int (f^{'} g) (x) d x \\ Proof: \\ (f g)^{'} = f^{'} g + f g^{'} \\ ⟹ \int (f g)^{'} (x) d x = \int (f^{'} g + f g^{'}) (x) d x = \int (f^{'} g) (x) d x + \int (f g^{'}) (x) d x \\ ⟹ f g = \int (f^{'} g) (x) d x - \int (f g^{'}) (x) d x ⟹ \int (f g^{'}) (x) d x = f g - \int (f^{'} g) (x) d x \end{matrix}

\begin{matrix} 1. Product of functions, when one function is an obvious derivative \\ with known primitive \\ \int x \sin x d x = \int x (- \cos x)^{'} d x = - x \cos x - \int - \cos x d x = - x \cos x + \sin x + C \\ Usuall, choosing polynomial as f is profitable \end{matrix}

\begin{matrix} \int \ln (x) d x = \int \ln (x) x^{'} d x = x \ln x - \int (\ln x)^{'} x d x = x \ln x - \int 1 d x = x \ln x - x + C \end{matrix}

\begin{matrix} \int \sin (\ln x) d x = \int \sin (\ln x) x^{'} d x = x \sin (\ln x) - \int \cos (\ln x) d x \\ \int \cos (\ln x) d x = \int \cos (\ln x) x^{'} d x = x \cos (\ln x) + \int \sin (\ln x) d x \\ ⟹ 2 \int \cos (\ln x) d x = x \sin (\ln x) + x \cos (\ln x) \\ ⟹ \int \cos (\ln x) d x = \frac{x \sin (\ln x) + x \cos (\ln x)}{2} + C \\ ⟹ \int \sin (\ln x) d x = \frac{x \sin (\ln x) - x \cos (\ln x)}{2} + C \end{matrix}