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.