Properties of Integration
$$ 1. \ \int f(x)dx=F(x)+c $$
$$ 2. \ \int af(x)dx=a \int f(x)dx $$
$$ 3. \ \int \left[ f(x)+g(x) \right] dx= \int f(x)dx + \int g(x)dx $$
$$ 4. \ \int \left[ f(x)-g(x) \right] dx= \int f(x)dx – \int g(x)dx $$
$$ 5. \ \int u \cdot v \ dx=u \int v \ dx – \int \left[ \left( \frac {du}{dx} \right) \left( \int v \ dx \right) \right] dx $$
$$ 6. \ \int f(ax)dx= \frac 1a F(ax)dx $$
$$ 7. \ \int \frac {f'(x)}{f(x)}dx= \log f(x)+c $$
$$ 8. \ \int f'(x)f(x)dx= \frac 12 f^2(x)+c $$
$$ 9. \ \int f(x)g'(x)dx= f(x)g(x)- \int g(x)f'(x)dx $$