Integrals are antiderivatives; so any derivative formula can be converted into an integral formula. In particular, we will use facts from Differentiation Techniques, including the following. In each case, we can add a constant term.
Basic Integrals
To prove the log integral, we note that
Trig functions
To prove the tan x integral, we note that if cos x > 0 then
While if cos x < 0 then
To prove the sec x integral, we note that if tan x + sec x > 0 , then
While if tan x + sec x < 0 , then
Arc trig antiderivatives
Based on the derivative rules for arc trig functions from Differentiation Techniques, we have
Arc hyperbolic antiderivatives
Based on the derivative rules for arc hyperbolic functions in Differentiation Techniques, we have
We now explore various techniques that are useful in simplifying many integrals. We begin with two more rules that carry over from differentiation.
Sum/Difference Rule
Constant Multiplier Rule
Integration by Substitution
We substitute u = u(x), and so du = u′dx. Thus
The definite integral version is
Example 1:
If a ≠ 0, then set u = ax+b, and so du = adx. It now follows that
When a ≠ 0 and c ≠ -1
When a ≠ 0 and c = -1
Of course, when a = 0, then
Example 2:
Set u = cos x. Then du = – sin x dx. Thus, dx = – du/sin x, and so
Example 3:
Using trig identities described in Differentiation Techniques, we obtain
More integration techniques
For more techniques, click on either of the following options
References
Bourne, M. (2026) Methods of integration
https://www.intmath.com/methods-integration/
Wikipedia (2026) Integration by substitution
https://en.wikipedia.org/wiki/Integration_by_substitution
Wikipedia (2026) Partial fraction decomposition
https://en.wikipedia.org/wiki/Partial_fraction_decomposition