Q.Evaluate: integral of 1 / sqrt(x - 1) dx.

Integration is, physically, the process of finding the AREA under a curve $y=f(x)$ plotted against $x$, and is the reverse operation of differentiation. The idea is built up by dividing the region under a curve, between $x=a$ and $x=b$, into a large number of thin vertical strips, approximating each strip as a rectangle, and summing their areas; as the number of strips is taken to infinity (each strip infinitesimally thin), this sum converges to the exact area, called the definite integral of $f(x)$ from $a$ to $b$, written $\int_a^b f(x)\,dx$.

The indefinite integral $F(x)=\int f(x)\,dx$ (with no limits) is the function whose derivative gives back $f(x)$, i.e. $f(x)=\frac{d}{dx}(F(x))$ — this is what makes integration the reverse of differentiation. A definite integral is then evaluated as $\int_a^b f(x)\,dx = F(b)-F(a)$. …