Integration
The Fundamental Thoery of Calculus #
Part 1 #
The First Fundamental Theorem of Calculus:
$$f(x) = \frac{d}{dx} \int_a^x f(x)dt $$
The order can be reversed:
$$f(x) = \frac{d}{dx} \int_a^x f(x)dt = - \frac{d}{dx} \int_x^a f(x)dt $$
Corollary (of the First Fundamental Theorem of Calculus): $$ \int^{g(x)}_{h(x)} f(t) dt = f[g(x)]g'(x) - f[h(x)]h'(x) $$
Part 2 #
$$\int_a^b{f(x)}dx = [F(x)]_a^b = F(a) - F(b)$$
Integrals of symetric functions #
- Integrals of odd functions: $\int_{-a}^a = 0$
- Integrals of even functions: $\int_{-a}^a = 2 \int_0^a$
Integration techniques #
Substitution rule #
$$ \int_a^b f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(u) du $$
- Find an $u$
- Calculate $du$ and express $dx$ in $du$
- Substitute $d$ and $dx$ and solve the integral by using the antiderevative of $u$
Integration by parts #
$$\int u(x)v'(x) dx = [u(x)v(x)] - \int v(x)u'(x) dx$$
Trigonometric substitution #
The following substitutions can be used to solve integrals with square roots:
| Expression | Substitution |
|---|---|
| $\sqrt{a^2 - x^2}$ | $x = a \sin{\theta}$ |
| $\sqrt{a^2 + x^2}$ | $x = a \tan{\theta}$ |
| $\sqrt{x^2 - a^2}$ | $x = a \sec{\theta}$ |
Method:
- Express $x$, and $dx$ in $\theta$.
- Substitute $x$ and $dx$ with $\theta$
- Substitute limits of definite integral with equivalent.
Partial fractions #
For integrating rational functions.
- If the degree of the nominator is greater than the degree of the denominator peform long division such that the degree of the nominator becomes less than the degree of the denominator.
- Factor the denominator as far as possible
- Express the function as a sum of partial fractions (4 cases).
Sometimes you need to use:
$$ \int \frac{1}{x^2 + a^2} dx = \frac{1}{a}\tan^{-1}{\Big(\frac{x}{a}\Big)} + C $$