Integration reverses differentiation. An indefinite integral gives a family of functions; a known point determines the constant.
\(\int \mathrm{f}(x)\,\mathrm{d}x=\mathrm{F}(x)+C,\qquad \mathrm{F}'(x)=\mathrm{f}(x)\).
For \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=6x-4\) and \(y=5\) when \(x=2\):
\(\begin{aligned}y&=3x^2-4x+C\\5&=12-8+C\\C&=1\\y&=3x^2-4x+1.\end{aligned}\)
Differentiate the result to check the derivative, then substitute the given point to check the constant.
If a rate is given as a function of time, its definite integral gives the net change in the quantity. Add the initial quantity to obtain its final value.
\(Q(b)=Q(a)+\int_a^b r(t)\,\mathrm{d}t\), where \(r(t)=\dfrac{\mathrm{d}Q}{\mathrm{d}t}\).
A tank contains \(12\) litres at \(t=0\), with net inflow \(r(t)=3t+2\) litres per minute. After \(2\) minutes, \(Q(2)=12+\left[\dfrac{3t^2}{2}+2t\right]_0^2=22\) litres.
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