Integration

Integral as (Signed) Area

- If $f$ is continuous and positive on $[a,b]$, $\int_a^b f(t)\,dt$ is the area under the graph between $x=a$ and $x=b$.

- If $f$ changes sign, the integral is the algebraic sum: positive areas where $f>0$, negative where $f<0$.

Basic Properties

- Zero interval: $\int_a^a f(t)\,dt = 0.$

- Orientation: $\int_a^b f(t)\,dt = -\int_b^a f(t)\,dt.$

- Chasles’ relation: for $c\in[a,b]$, $\int_a^b f(t)\,dt = \int_a^c f(t)\,dt + \int_c^b f(t)\,dt.$

- Linearity: for continuous $f,g$ on $[a,b]$ and $\lambda\in\mathbb{R}$,

- $\int_a^b (f+g)(t)\,dt = \int_a^b f(t)\,dt + \int_a^b g(t)\,dt,$

- $\int_a^b \lambda f(t)\,dt = \lambda \int_a^b f(t)\,dt.$

- Positivity / comparison:

- If $f\ge 0$ on $[a,b]$, then $\int_a^b f(t)\,dt \ge 0.$

- If $f\le g$ on $[a,b]$, then $\int_a^b f(t)\,dt \le \int_a^b g(t)\,dt.$

#Primitive (Antiderivative)

- A primitive of $f$ on an interval $I$ is a function $F$ with $F'(x)=f(x)$ for all $x\in I$.

- If $F$ is one primitive, then all primitives of $f$ are $F+C$ with $C\in\mathbb{R}$.

- The function $F(x)=\int_a^x f(t)\,dt$ is a primitive of $f$ on $[a,b]$ and satisfies $F(a)=0$ (distinguished primitive).

- Fundamental theorem of calculus: for continuous $f$ on $[a,b]$ and any primitive $F$, $\int_a^b f(t)\,dt = F(b) - F(a) = [F(x)]_a^b.$

Usual Primitives

- Power functions ($\varepsilon\neq -1$): $\int x^\varepsilon\,dx = \frac{x^{\varepsilon+1}}{\varepsilon+1} + C.$

- Logarithm case: $\int \frac{1}{x}\,dx = \ln|x| + C.$

- Exponential: $\int e^x\,dx = e^x + C.$

- Trigonometric: $\int \cos x\,dx = \sin x + C,\qquad \int \sin x\,dx = -\cos x + C.$

- Arctangent: $\int \frac{1}{1+x^2}\,dx = \arctan x + C.$

Integration by Parts

- Based on $(fg)' = f'g + fg'.$

- Primitive form: $\int f'(x)g(x)\,dx = f(x)g(x) - \int f(x)g'(x)\,dx.$

- Definite integral form: $\int_a^b f'(x)g(x)\,dx = [f(x)g(x)]_a^b - \int_a^b f(x)g'(x)\,dx.$

- Typical strategy: choose $g$ to simplify when differentiated and $f'$ to be easily integrable.

# Example: Primitive of $\ln x$

- On $(0,+\infty)$: $\int \ln x\,dx = x\ln x - x + C,$ via integration by parts with $f'(x)=1$, $g(x)=\ln x$.

Change of Variables (Substitution)

- General theorem: if $u:[a,b]\to I$ is $\mathcal{C}^1$ with continuous derivative and $f$ is continuous on $I$, then $\int_a^b f(u(t))u'(t)\,dt = \int_{u(a)}^{u(b)} f(x)\,dx.$

- Careful: the bounds must also be transformed: $a,b \mapsto u(a),u(b)$.

- Primitive version: $\int f(u(x))u'(x)\,dx = F(u(x)) + C,$ where $F$ is a primitive of $f$.

Translation / Affine Change of Variable

- Translation: $\int_a^b f(x+x_0)\,dx = \int_{a+x_0}^{b+x_0} f(t)\,dt.$

- General affine change $u(x)=px+x_0$ ($p\neq0$):

- Primitive: $\int f(px+x_0)\,dx = \frac{1}{p}F(px+x_0)+C,$ where $F' = f$.

- Definite integral: $\int_a^b f(px+x_0)\,dx = \frac{1}{p}\int_{pa+x_0}^{pb+x_0} f(x)\,dx.$

“Physicist’s Method” Notation

- Write $t = u(x)$, compute $\dfrac{dt}{dx}$, rearrange symbolically as $dx = \cdots\,dt$ to do substitution quickly.

- Same idea as rigorous change of variable, but with differential notation.

Example Types

- Integral of $t e^{-t}$ on $[0,T]$ by parts; shows a limit as $T\to+\infty$ equal to $1$ (generalized integral): $\int_0^T t e^{-t}\,dt.$

- Substitution examples such as $\int_1^9 \frac{1}{1+\sqrt{x}}\,dx,$ with $t=\sqrt{x}$.

- Completing the square and substitution for rational functions like $\int \frac{dx}{x^2+4x+8}.$

Applied Example – River Discharge

- A discharge function $q(t)$ models water flow rate (e.g. in millions of $\text{m}^3$ per day).

- Total volume over $[0,T]$ is $v(T) = \int_0^T q(t)\,dt.$

- In the example, $q(t)$ is given in terms of a logistic-type expression, and recognizing it as $q(t) = c\,\frac{-u'(t)}{u(t)^2}$ allows finding a primitive analytically.