P4 Integration Flashcards
integrate x^n dx
1 / (n+1) x X^(n+1) + c
integrate e^x dx
e^x + c
integrate 1 / x dx
ln(x) + c
integrate cos(x) dx
sin(x) + c
integrate sin(x) dx
- cos(x) + c
integrate sec^2(x) dx
tan(x) + c
integrate cosec(x) cot(x) dx
- cosec(x) + c
integrate cosec^2(x) dx
- cot(x) + c
integrate sec(x) tan(x) dx
sec(x) + c
Integrating by reverse chain rule
integrate f’ (ax + b) dx = 1/a f (ax + b) + c
Integrating using trig identities
some expressions, such as sin^2(x) and sin(x)cos(x) cannot be integrated directly, but we can use our trig identities to replace them with expressions we can easily integrate
Integrating using partial identities
some expressions can be split into partial fractions which allows us to differentiate some expressions with more complicated denominators
- after finding the partial fractions, use ln like for ln | x+1 |
Integrating using “consider and scale”
There’s certain more complicated expressions which look like the result of having applied the chain rule
the process is then simply
1) consider some expression that will differentiate to something similar to it
2) differentiate, and adjust for any scale difference
Integrating by substitution
For some integrations involving a complicated expression, we can make a substitution to turn it into an equivalent integration that is simpler.
- can’t use reverse chain rule on these expressions
Integrating by parts
integrated u dv/dx dx = uv - integrated v du/dx dx
