Further Calculus (HL)

Higher Level calculus extends the toolkit to integrals that resist the basic rules, to rates that are linked through time, to solids and to infinite series. This lesson surveys the core HL techniques.

Integration by substitution. This reverses the chain rule. To integrate $\int f(g(x))\,g'(x)\,dx$, set $u = g(x)$ so that $du = g'(x)\,dx$; the integral becomes $\int f(u)\,du$, which is hopefully simpler. Look for an inner function whose derivative also appears (up to a constant) in the integrand.

Worked example (substitution). Evaluate $\int 2x(x^2 + 1)^5\,dx$. Let $u = x^2 + 1$, so $du = 2x\,dx$. The integral becomes $\int u^5\,du = \frac{u^6}{6} + C = \frac{(x^2+1)^6}{6} + C$. For definite integrals, either convert the limits to $u$-values or convert back to $x$ before substituting the or