Integration Flashcards
Basic integration
β« ππ₯βΏ ππ₯ =
[ππ₯βΏβΊΒΉ] / [π+1] + π , π β β1
Properties of indefinite integral (3)
β«[π(π₯) + π(π₯)] ππ₯ = β« π(π₯) ππ₯ + β« π(π₯) ππ₯
β«[π(π₯) β π(π₯)] ππ₯ = β« π(π₯) ππ₯ β β« π(π₯) ππ₯
β« ππ(π₯) ππ₯ = π β« π(π₯) ππ₯ π = constant
The definition of a definite integral
β«α΅β π(π₯) ππ₯ = [πΉ(π₯)]α΅β = πΉ(π) β πΉ(π)
Integral properties
β«α΅β π(π₯) ππ₯ = β β«α΅α΅¦ π(π₯) ππ₯
β«α΅β π(π₯) ππ₯ = 0
Integration of exponential functions πΛ£
β«eΛ£ dx = eΛ£ + c
β«eα΅Λ£ dx = 1/k eα΅Λ£ + c , k β 0
Integration of reciprocal 1/ππ₯+b
β«1/π₯ ππ₯ = ππ|π₯| + π , π₯ β 0
Take any constants out in their form
eg
5/x becomes 5ln|x|+ c
1/5x becomes 1/5ln|x|+ c
Special cases for trig integration
β«sinΒ²(x) dx = x/2 β 1/4 sin(2x) + c
β«cosΒ²(x) dx = x/2 + 1/4 sin(2x) + c
The trapezium rule
Area of Trapezium = ([π+π]/2) Γ β
The trapezium rule with π intervals states that
β«α΅β π(π₯) ππ₯ β h/2 (yβ+2yβ+2yβ+ β¦ +2yβββ+yβ)
where h = b-a/n
Integrating quotient t ππβ²(π₯)/π(π₯)
β«fβ(x)/f(x) dx = ln|f(x)| + c
β«kfβ(x)/f(x) dx = k ln|f(x)| + c
Integrating chain rule
β« gβ(x) x fβ(g(x) = f[g(x) + c
Integration by substitution
Steps:
1. Choose correct substitution (of u)
2. Change everything in terms of new variable and calculate new integral
3. Change back to original variable
Integration of rational functions
When finding integrals such as
ax + b / cx + d and other expressions in fractional form it is sometimes best to divide using algebra long division first
remaining is the numerator
successful division is added
Integration with partial fractions
When given an equation that can be turned into a partial fraction, convert it and then integrate using standard integration procedures
Integration by parts
Recall that if u and v are differentiable functions of x, uv = uvβ + vβu
β«uvβ = uv - β«vuβ
Note: the goal of this procedure is to end up with a single term or terms that can be integrated after the β«
