Antiderivative Rules and Methods Flashcards
What do we call F(x) such that dF/dx = f(x)?
Antiderivative
Power Rule: ∫x^n dx = ?
[x^(n + 1)]/(n + 1) + C
When does the Power Rule not work?
When the exponent is -1
Constant Multiple Rule:
∫a*f(x) dx = ?
a∫f(x) dx
Sum/Difference Rule:
∫f(x) + g(x) dx = ?
∫f(x) dx + ∫g(x) dx
Can you use a product or quotient rule for integrals?
No
For constant a:
∫a dx = ?
ax + C
∫e^x dx = ?
e^x + C
∫1/x dx = ?
ln|x| + C
∫cos(x) dx = ?
sin(x) + C
∫sin(x) dx = ?
-cos(x) + C
∫x^-1 dx = ?
∫1/x dx = ln|x| + C
Which method of evaluating antiderivatives undoes the chain rule?
U-substitution
Formula for integrating by parts
∫u dv = uv - ∫v du
When integrating by parts, what is the requirement for assigning one part as dv?
It must be something you can integrate
When integrating by parts, what is the requirement for assigning one part as u?
It must get nicer when you take the derivative
What two things are required for u-substitution?
The function u and its derivative behind the integrand ∫
For ∫(linear)f(linear), what method do you use?
Parts, then u-sub
