Q.If $f''(x) = 12x-6$ and $f(1)=30,\ f'(1)=5$, find $f(x)$.

Linearity of integration. For a constant $k$ and integrable functions $f_1,f_2,\dots$:

Combined and extended: $\int\big(k_1 f_1 \pm k_2 f_2 \pm \cdots \pm k_n f_n\big)\,dx = k_1\!\int f_1\,dx \pm \cdots \pm k_n\!\int f_n\,dx$ — the integral of a linear combination is the linear combination of the integrals.

Simple applications. Integration recovers a function from its rate of change. If the derivative $\frac{dy}{dx}=f'(x)$ is known, integrating gives $y=f(x)+c$, and an initial condition fixes $c$. In kinematics, with the sign conventions of motion:

• acceleration $a=\dfrac{dv}{dt}$, so $v=\int a\,dt$;
• velocity $v=\dfrac{ds}{dt}$, so $s=\int v\,dt$. …