Quick questions on Integration techniques: antiderivatives, substitution, definite integrals and the FTC

If the argument is linear, divide by the coefficient.

The substitution rule reverses the chain rule. To evaluate $\int f(g(x)) g'(x) \, dx$, set $u = g(x)$, so $du = g'(x) \, dx$. The integral becomes $\int f(u) \, du$, which you evaluate, then substitute back.

The whole skill of substitution is recognising that the integrand contains a function and (a multiple of) its derivative. Scan the integrand for an inner function $u = g(x)$ whose derivative $g'(x)$ also appears, possibly up to a constant factor. Good signals include a power of a bracket multiplied by the bracket's derivative, a function inside a root, or a fraction whose numerator is the derivative of the denominator. If no such pair appears, substitution will not help and you should look for a standard form instead.

Geometrically, $\int_a^b f(x)\,dx$ is the signed area between the curve and the $x$-axis: regions above the axis count positive, regions below count negative. This is why a definite integral can be zero even when the curve is non-zero, as when an odd function is integrated over a symmetric interval. To find a genuine geometric area where the curve crosses the axis, split the integral at the crossing points and add the magnitudes, exactly as you split a motion problem at the times when velocity is zero.

The single most useful picture in this whole topic is the definite integral as a signed area. Below it is built up one stage at a time on a curve that is above the axis on the left part of the interval and below it on the right, so you can see exactly where the sign comes from.

Two properties simplify many definite integrals: $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$ (reversing the limits flips the sign), and $\int_a^b f + \int_b^c f = \int_a^c f$ (adjacent intervals combine). For an even function $\int_{-a}^{a} f = 2\int_0^a f$, and for an odd function $\int_{-a}^{a} f = 0$. Quoting the odd-function rule on a symmetric interval can turn a page of working into a one-line "the integral is $0$ by symmetry", which markers accept and which removes the chance of an arithmetic slip.