Q.Evaluate: ∫ (1/√x) dx

Mathematical Tools: Differentiation and Integration

Physics is written in the language of change, and calculus is the grammar. Before tackling motion, fields or thermodynamics, one needs two tools: differentiation (finding a rate of change) and integration (adding up continuously varying quantities).

Differentiation — the rate of change

The derivative $\dfrac{dy}{dx}$ measures how fast $y$ changes with $x$. From first principles,

Standard results used constantly in physics:

In physics: velocity is the derivative of position, $v = dx/dt$, and acceleration the derivative of velocity, $a = dv/dt$. A slope on a graph is a derivative.

Integration — the continuous sum

Integration is the reverse operation: given a rate, recover the total. The indefinite integral is the antiderivative,

while the definite integral gives the accumulated amount between limits, geometrically the area under the curve:

$\int_a^b f(x)\,dx = F(b) - F(a)$ …