# 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:

ExpressionSubstitution
$\sqrt{a^2 - x^2}$$x = a \sin{\theta} \sqrt{a^2 + x^2}$$x = a \tan{\theta}$