<s>[INST] Find the value $\int_{0}^{1}f(x) \,\mathrm dx$ without words 
Let $f(x)=ax^3+bx^2+cx+d$ such that: $$f(0)=1,\quad f(0.5)=5, \quad f(1)=15$$
Find: $$\int_{0}^{1}f(x) \,\mathrm dx$$

It is said that it can be solved without words.
 [/INST] $f(0) = 1\implies d = 1, f(0.5) = 5 \implies \dfrac{a}{8} +\dfrac{b}{4} + \dfrac{c}{2} = 4, f(1) = 15 \implies 2a+2b+2c = 28\implies \displaystyle \int_{0}^{1} f(x)dx = \dfrac{a}{4}+\dfrac{b}{3}+\dfrac{c}{2}+d= \dfrac{3a+4b+6c}{12}+1=\dfrac{32+28}{12}+1 = 6$
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