Integration Flashcards
What is integration the reverse of?
Differentiation.
What is the integral of xⁿ (where n ≠ -1)?
∫xⁿ dx = xⁿ⁺¹ / (n+1) + C
What is ∫1/x dx?
ln|x| + C
What is ∫eˣ dx?
eˣ + C
What is ∫cos(x) dx?
sin(x) + C
What is ∫sin(x) dx?
-cos(x) + C
What is ∫ln(x) dx?
xln(x) - x + C
What does the constant of integration ‘C’ represent?
An unknown constant added when finding indefinite integrals.
What is a definite integral?
An integral with upper and lower limits which gives a numerical value.
How do you calculate a definite integral?
Find the antiderivative, then subtract f(a) from f(b): ∫[a to b] f(x) dx = F(b) - F(a).
What does a definite integral represent graphically?
The signed area under the curve between the limits.
What is the trapezium rule used for?
Numerical approximation of definite integrals using trapeziums.
What is integration by substitution?
Replacing a complex expression with a simpler variable to integrate more easily.
What is integration by parts?
∫u dv = uv - ∫v du
What is ∫x eˣ dx (by parts)?
x eˣ - ∫eˣ dx = x eˣ - eˣ + C
What is the integral of a product of functions?
Use integration by parts.
How do you recognise a substitution integral?
Look for a composite function where inner derivative also appears.
What is the area between two curves y₁ and y₂ from a to b?
∫[a to b] (y₁ - y₂) dx
What is ∫sec²x dx?
tan(x) + C
How can integration be used in kinematics?
To find displacement from velocity or velocity from acceleration.
