Fundamental Theorem of Calculus

Question

Let

∫²₀ 𝑓(𝑥)𝑑𝑥=13, ,∫³₀ 𝑓(𝑥)𝑑𝑥=1,

∫²₀ 𝑔(𝑥)𝑑𝑥=−12, ∫³₂ 𝑔(𝑥)𝑑𝑥=−10,

Use these values to evaluate the given definite integrals.

a) ∫²₀(𝑓(𝑥)+𝑔(𝑥))𝑑𝑥=

b) ∫³₀(𝑓(𝑥)−𝑔(𝑥))𝑑𝑥=

c) ∫³₂(3𝑓(𝑥)+2𝑔(𝑥))𝑑𝑥=

d) Find the value 𝑎a such that  ∫³₀ (a𝑓(𝑥)+𝑔(𝑥)) 𝑑𝑥 = 0

Answer 1

The trick here is that ∫³₀ g(x)dx needs to be derived by adding ∫²₀ 𝑔(𝑥)𝑑𝑥 and ∫³₂ 𝑔(𝑥)𝑑x.

The rest is pretty straight forward using integral addition, subtraction a multiplication rules.