Integration Techniques Cheat Sheet

A printable reference covering substitution, integration by parts, trigonometric integrals, partial fractions, and improper integrals for grades 11-12.

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Integration techniques help students evaluate antiderivatives that are not handled by basic power, exponential, logarithmic, or trigonometric rules alone. This cheat sheet summarizes the main strategies used in Grade 11–12 calculus, including substitution, integration by parts, trigonometric methods, partial fractions, and improper integrals. Students need these tools to recognize patterns, choose efficient methods, and check answers by differentiation. A compact reference makes it easier to compare techniques while solving mixed integration problems. The most important idea is to match the integrand to a structure you can simplify. Substitution reverses the chain rule, while integration by parts reverses the product rule. Trigonometric identities and substitutions transform difficult radicals or powers into standard forms. Partial fractions break rational functions into simpler terms, and improper integrals use limits to decide whether an infinite interval or discontinuous integral converges.

Key Facts

The substitution rule is $\int f(g(x))g'(x)\,dx = \int f(u)\,du$ where $u = g(x)$ and $du = g'(x)\,dx$ .
Integration by parts is $\int u\,dv = uv - \int v\,du$ , which comes from the product rule.
For definite substitution, change the limits using $u = g(a)$ and $u = g(b)$ so that $\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$ .
A common trigonometric identity for integrals is $\sin^2 x + \cos^2 x = 1$ , and half-angle forms include $\sin^2 x = \frac{1 - \cos 2x}{2}$ and $\cos^2 x = \frac{1 + \cos 2x}{2}$ .
For radicals, common trigonometric substitutions include $x = a\sin \theta$ for $\sqrt{a^2 - x^2}$ , $x = a\tan \theta$ for $\sqrt{a^2 + x^2}$ , and $x = a\sec \theta$ for $\sqrt{x^2 - a^2}$ .
Partial fractions apply when a rational function $\frac{P(x)}{Q(x)}$ has $\deg P(x) < \deg Q(x)$ , and long division is needed first if $\deg P(x) \ge \deg Q(x)$ .
An improper integral over an infinite interval is defined by a limit, such as $\int_a^{\infty} f(x)\,dx = \lim_{b\to\infty}\int_a^b f(x)\,dx$ .
An antiderivative answer should be checked by differentiating: if $F'(x) = f(x)$ , then $\int f(x)\,dx = F(x) + C$ .

Vocabulary

Antiderivative: An antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$ .
Substitution: Substitution rewrites an integral using $u = g(x)$ and $du = g'(x)\,dx$ to simplify a composite expression.
Integration by Parts: Integration by parts is the method $\int u\,dv = uv - \int v\,du$ used when an integrand looks like a product of functions.
Partial Fractions: Partial fractions decompose a rational expression into simpler fractions that are easier to integrate.
Improper Integral: An improper integral is an integral with an infinite limit of integration or an integrand that is discontinuous on the interval.
Convergence: An improper integral converges if the limit defining the integral exists and equals a finite number.

Common Mistakes to Avoid

Forgetting $du$ in substitution is wrong because changing variables requires replacing both the expression and the differential, not just part of the integrand.
Choosing poor parts in $\int u\,dv$ is wrong because it can make $\int v\,du$ harder than the original integral instead of simpler.
Not changing limits in a definite $u$ -substitution is wrong because the new variable has different endpoint values from the original variable.
Using partial fractions before proper division is wrong when $\deg P(x) \ge \deg Q(x)$ because the rational expression must first be rewritten as a polynomial plus a proper fraction.
Treating an improper integral like an ordinary definite integral is wrong because infinite bounds or discontinuities must be handled with limits before deciding convergence.

Practice Questions

1 Evaluate $\int 2x\sqrt{x^2 + 5}\,dx$ using substitution.
2 Evaluate $\int x e^{3x}\,dx$ using integration by parts.
3 Decompose and evaluate $\int \frac{5x + 1}{x^2 - x - 2}\,dx$ using partial fractions.
4 Explain which technique you would try first for $\int x\sqrt{9 - x^2}\,dx$ and why that method matches the structure of the integrand.