# Properties of Indefinite Integrals

• Functions: $$f$$, $$g$$, $$u$$, $$v$$, $$F$$
Independent variables: $$x$$, $$t$$
Constant real numbers: $$C$$, $$a$$, $$b$$, $$k$$
1. An antiderivative of a function $$y = f\left( x \right)$$ defined on some interval $$\left( {a,b} \right)$$ is called any function $$F\left( x \right)$$ whose derivative at any point of this interval is equal to $$f\left( x \right)$$:
$$F’\left( x \right) = f\left( x \right)$$.
If $$F\left( x \right)$$ is an antiderivative of $$f\left( x \right)$$, then the function of the form $$F\left( x \right) + C$$, where $$C$$ is an arbitrary constant, is also an antiderivative of $$f\left( x \right)$$.
2. The indefinite integral of a function $$f\left( x \right)$$ is the collection of all antiderivatives for this function:
$$\int {f\left( x \right)dx} = F\left( x \right) + C,$$ if $$F’\left( x \right) = f\left( x \right).$$
3. The derivative of the indefinite integral is equal to the integrand:
$${\left( {\int {f\left( x \right)dx} } \right)^\prime } = f\left( x \right).$$
4. The indefinite integral of the sum of two functions is equal to the sum of the integrals:
$$\int {\left[ {f\left( x \right) + g\left( x \right)} \right]dx} =$$ $$\int {f\left( x \right)dx} + \int {g\left( x \right)dx} .$$
5. The indefinite integral of the difference of two functions is equal to the difference of the integrals:
$$\int {\left[ {f\left( x \right) – g\left( x \right)} \right]dx} =$$ $$\int {f\left( x \right)dx} – \int {g\left( x \right)dx} .$$
6. A constant factor can be moved across the integral sign:
$$\int {kf\left( x \right)dx} = k\int {f\left( x \right)dx} .$$
7. $$\int {f\left( {ax} \right)dx} =$$ $${\large\frac{1}{a}\normalsize} F\left( {ax} \right) + C$$
8. $$\int {f\left( {ax + b} \right)dx} =$$ $${\large\frac{1}{a}\normalsize} F\left( {ax + b} \right) + C$$
9. $$\int {f\left( x \right)f’\left( x \right)dx} =$$ $${\large\frac{1}{2}\normalsize} {f^2}\left( x \right) + C$$
10. $$\int {{\large\frac{{f’\left( x \right)}}{{f\left( x \right)}}\normalsize} dx} =$$ $$\ln \left| {f\left( x \right)} \right| + C$$
11. Integration by substitution
$$\int {f\left( x \right)dx} = \int {f\left( {u\left( t \right)} \right)u’\left( t \right)dt} ,$$ if $$x = u\left( t \right).$$
12. Integration by parts
$$\int {udv} = uv – \int {vdu} ,$$
where $$u\left( x \right)$$, $$v\left( x \right)$$ are differentiable functions.