Integration Techniques

u-substitution and integration by parts.

The idea

An antiderivative is found by recognition: each differentiation rule, read in reverse, tells us which integrands it produces. Two rules, reversed, account for most integrals met in practice.

Theorem (Substitution rule).

If $g$ is differentiable and $f$ is continuous, then with $u = g(x)$ and $du = g'(x)\,dx$, $\int f(g(x))\,g'(x)\,dx = \int f(u)\,du.$

Substitution reverses the chain rule. That rule turns $F(g(x))$ into $f(g(x))\,g'(x)$, so an integrand of that shape — a function composed with an inner function, multiplied by the inner function's derivative — must have come from $F(g(x))$. Naming the inner function $u$ collapses the integral to one in a single variable with no composition left. The skill is spotting an inner function whose derivative already appears in the integrand, up to a constant factor.

Theorem (Integration by parts).

If $u$ and $v$ are differentiable functions of $x$, then $\int u\,dv = uv - \int v\,du.$

Integration by parts reverses the product rule. Differentiating $uv$ produces $u\,dv + v\,du$; integrating that identity and moving one piece across gives the formula. It does not evaluate the integral; it trades it for another. The choice of $u$ and $dv$ therefore decides everything: pick them so that the remaining integral is easier, which usually means letting $u$ be the factor that simplifies when differentiated.

Ways to work on it

Not sure where to start? Take the ten-question placement test.