Integration Techniques Cheat Sheet

Quick answer

Choose an integration technique by matching the integrand's structure: substitution for a composite pattern, parts for products, and partial fractions for rational functions.

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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.

Understanding Integration Techniques

Choosing a method starts with inspecting the whole expression, not just spotting a familiar function. Look for an inner expression whose derivative is present nearby, perhaps with a constant factor. For example, a factor of two x strongly suggests using x squared plus one as the new variable.

If the needed derivative differs only by a number, multiply and divide by that number before starting. This small adjustment is legal because it does not change the value of the integrand.

Write the replacement clearly, then rewrite every part of the integral in the new variable. Mixing old and new variables is one of the most common substitution errors.

Integration by parts needs a deliberate choice for the part called u. A useful classroom guideline puts logarithms first, then inverse trigonometric functions, algebraic powers, trigonometric functions, and exponentials. This often makes repeated differentiation simpler.

For instance, differentiating a polynomial eventually gives zero, while integrating an exponential keeps it manageable. The method can fail to improve an integral when the choices are poor. Sometimes the original integral appears again after one or two steps.

In that case, give the repeated integral a name, move it to the other side, and solve for it. This is an algebra step, not a new integration rule.

Trigonometric work is mostly about changing an expression into terms that have known antiderivatives. When powers of sine or cosine occur, first check whether one power is odd. You can save one factor and turn the remaining even power into the other function using an identity.

When both powers are even, half angle identities are usually more efficient. For rational functions, factor the denominator as far as possible before writing partial fractions. Different factors need different forms.

A repeated linear factor needs a separate fraction for each power. A quadratic factor that cannot be factored over the real numbers needs a linear numerator. Omitting one required term produces coefficients that cannot match the original fraction.

Improper integrals teach an important distinction between a function becoming small and an accumulated area staying finite. A positive function can approach zero while its total area grows without bound. This matters in models of cooling, radioactive decay, probability distributions, and long term costs.

Split an interval whenever there is a discontinuity inside it, then test each piece with its own limit. Never substitute infinity into an antiderivative as though it were an ordinary number.

As you practise, record why each technique was chosen and differentiate the final result. That habit reveals missing constants, sign errors, and incorrect algebra before they become hard to find.

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