Integration Techniques Selector and Practice

Math Calculus • Integrals

Written by STEM Calculators Team Published June 15, 2026 Updated June 15, 2026

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Theory

How to choose an integration technique

Choosing an integration technique means looking at the structure of the integrand. A good order is:

Check basic formulas.
Check for \(u\)-substitution.
Check for integration by parts.
Check for trigonometric identities.
Check for trigonometric substitution.
Check for partial fractions.
If no elementary method works, use a numerical method or special function.

The goal is not only to integrate, but to recognize why a method fits.

Basic rules

Always check basic rules first. They are the fastest.

\[ \begin{aligned} \int x^n\,dx &= \frac{x^{n+1}}{n+1}+C, \qquad n\ne -1 \end{aligned} \]

For \(n=-1\), use:

\[ \begin{aligned} \int \frac{1}{x}\,dx &= \ln|x|+C \end{aligned} \]

Other basic rules include:

\[ \begin{aligned} \int e^x\,dx&=e^x+C\\ \int \sin x\,dx&=-\cos x+C\\ \int \cos x\,dx&=\sin x+C\\ \int \frac{1}{1+x^2}\,dx&=\arctan x+C \end{aligned} \]

\(u\)-substitution

Use \(u\)-substitution when the integrand contains an inside function and its derivative.

The pattern is:

\[ \begin{aligned} \int f(g(x))g'(x)\,dx &= \int f(u)\,du \end{aligned} \]

Example:

\[ \begin{aligned} \int (2x+3)^5\,dx \end{aligned} \]

Let:

\[ \begin{aligned} u&=2x+3\\ du&=2\,dx \end{aligned} \]

Then:

\[ \begin{aligned} \int (2x+3)^5\,dx &= \frac{1}{2}\int u^5\,du\\ &= \frac{u^6}{12}+C\\ &= \frac{(2x+3)^6}{12}+C \end{aligned} \]

Logarithmic \(u\)-substitution

A common \(u\)-substitution pattern is:

\[ \begin{aligned} \int \frac{g'(x)}{g(x)}\,dx &= \ln|g(x)|+C \end{aligned} \]

Example:

\[ \begin{aligned} \int \frac{2x}{x^2+1}\,dx \end{aligned} \]

Let:

\[ \begin{aligned} u&=x^2+1\\ du&=2x\,dx \end{aligned} \]

Therefore:

\[ \begin{aligned} \int \frac{2x}{x^2+1}\,dx &= \int \frac{du}{u}\\ &= \ln|u|+C\\ &= \ln(x^2+1)+C \end{aligned} \]

Integration by parts

Integration by parts is useful for products. The formula is:

\[ \begin{aligned} \int u\,dv &= uv-\int v\,du \end{aligned} \]

A common guide for choosing \(u\) is LIATE:

Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential

Example:

\[ \begin{aligned} \int xe^x\,dx \end{aligned} \]

Choose:

\[ \begin{aligned} u=x, \qquad dv=e^x\,dx \end{aligned} \]

Then:

\[ \begin{aligned} du=dx, \qquad v=e^x \end{aligned} \]

So:

\[ \begin{aligned} \int xe^x\,dx &= xe^x-\int e^x\,dx\\ &= xe^x-e^x+C \end{aligned} \]

Trigonometric identities

Use trig identities when the integrand contains powers or products of trigonometric functions.

For odd powers of sine:

\[ \begin{aligned} \sin^3x &= \sin x(1-\cos^2x) \end{aligned} \]

Therefore:

\[ \begin{aligned} \int \sin^3x\,dx &= -\cos x+\frac{1}{3}\cos^3x+C \end{aligned} \]

For even powers, use power-reduction identities:

\[ \begin{aligned} \sin^2x&=\frac{1-\cos(2x)}{2}\\ \cos^2x&=\frac{1+\cos(2x)}{2} \end{aligned} \]

Trigonometric substitution

Use trigonometric substitution for radical expressions of special forms.

\[ \begin{array}{c|c} \text{Radical form} & \text{Substitution}\\ \hline \sqrt{a^2-x^2} & x=a\sin\theta\\ \sqrt{x^2-a^2} & x=a\sec\theta\\ \sqrt{x^2+a^2} & x=a\tan\theta \end{array} \]

Example:

\[ \begin{aligned} \int \frac{dx}{\sqrt{9-x^2}} \end{aligned} \]

Since \(9=3^2\), use:

\[ \begin{aligned} x=3\sin\theta \end{aligned} \]

The final result is:

\[ \begin{aligned} \int \frac{dx}{\sqrt{9-x^2}} &= \arcsin\left(\frac{x}{3}\right)+C \end{aligned} \]

Partial fractions

Use partial fractions for rational functions whose denominators can be factored.

Example:

\[ \begin{aligned} \int \frac{3x+2}{(x-1)(x+2)}\,dx \end{aligned} \]

Decompose:

\[ \begin{aligned} \frac{3x+2}{(x-1)(x+2)} &= \frac{A}{x-1} + \frac{B}{x+2} \end{aligned} \]

Solve for \(A\) and \(B\):

\[ \begin{aligned} 3x+2 &= A(x+2)+B(x-1) \end{aligned} \]

This gives:

\[ \begin{aligned} A=\frac{5}{3}, \qquad B=\frac{4}{3} \end{aligned} \]

Therefore:

\[ \begin{aligned} \int \frac{3x+2}{(x-1)(x+2)}\,dx &= \frac{5}{3}\ln|x-1| + \frac{4}{3}\ln|x+2| +C \end{aligned} \]

Numerical and special-function cases

Some integrals do not have elementary antiderivatives.

A famous example is:

\[ \begin{aligned} \int e^{-x^2}\,dx \end{aligned} \]

This integral is related to the error function, written \(\operatorname{erf}(x)\). For a definite integral, a numerical method such as Simpson's rule can give a very accurate estimate.

In practice, if no elementary technique works, switch from symbolic integration to numerical integration or a special-function representation.

Common mistakes

Using substitution for every integral: substitution works only when the inside derivative is present.
Using parts too early: check basic rules first.
Forgetting \(+C\): every indefinite integral needs a constant of integration.
Ignoring absolute value: logarithmic answers often require \(\ln|u|\).
Choosing the wrong \(u\): for integration by parts, choose the factor that becomes simpler after differentiation.
Trying elementary methods on non-elementary integrals: some integrals require special functions or numerical methods.
Forgetting back-substitution: after \(u\)-substitution or trig substitution, return to the original variable.

Frequently Asked Questions

How does the selector choose an integration technique?

It checks the structure of the integrand: simple formulas first, then u-substitution, products for integration by parts, trig patterns, radical patterns, rational functions, and finally numerical or special-function cases.

When should I use u-substitution?

Use u-substitution when the integrand contains a composition and the derivative of the inside function is also present, possibly up to a constant factor.

When should I use integration by parts?

Use integration by parts when the integrand is a product and one factor becomes simpler after differentiation, such as x exp(x), x sin(x), or ln(x).

When should I use trigonometric identities?

Use trig identities for powers or products of trigonometric functions, such as sin^3(x), cos^3(x), sin^2(x), or cos^2(x).

When should I use trigonometric substitution?

Use trig substitution for radical forms such as sqrt(a^2 - x^2), sqrt(x^2 - a^2), and sqrt(x^2 + a^2).

When should I use partial fractions?

Use partial fractions for rational functions where the denominator factors into simpler linear or irreducible quadratic factors and the numerator degree is smaller than the denominator degree.

What if no elementary method works?

For some integrals, such as exp(-x^2), no elementary antiderivative exists. Definite integrals can still be estimated numerically or expressed using special functions.

Does this replace a full computer algebra system?

No. It is a teaching and practice tool focused on common integration patterns and method selection.

Why does practice mode hide the solution?

Practice mode is designed to help students think through the method before seeing the final answer.

Can it handle definite integrals?

Yes. If a supported antiderivative is found, it evaluates the definite integral from the antiderivative. Otherwise, it attempts a numerical estimate.

Integration Techniques Selector and Practice

Integral to analyze

Output and practice settings

Quick examples

Technique decision map

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Frequently Asked Questions

Integral to analyze

Output and practice settings

Quick examples

Technique decision map

Rate this calculator

Frequently Asked Questions

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