To find an antiderivative, first simplify the function, then check whether a basic rule fits (power, \(e^x\), \(\frac1x\), sin, cos). If there is an inside function whose derivative is also present, use u-substitution. For a product of different kinds of functions, use integration by parts. For a fraction of polynomials, use partial fractions. Always check by differentiating.
Most textbooks teach integration methods one chapter at a time. The hard part comes later, when a test gives you a mixed problem and you have to decide which method to use. This guide is a decision checklist: eight questions to ask, in order, with an example for each. Work down the list and stop at the first “yes”.
The 8-Question Checklist
| # | Ask yourself… | If yes, use… |
|---|---|---|
| 1 | Can I simplify it first? | Algebra: expand, split, rewrite powers |
| 2 | Is every piece in the basic table? | The basic rules |
| 3 | Is there an inside function with its derivative nearby? | U-substitution |
| 4 | Is it a product of two different kinds of functions? | Integration by parts |
| 5 | Is it a fraction of two polynomials? | Long division / partial fractions |
| 6 | Is it a power of sin, cos, tan or sec? | Trig identities |
| 7 | Is there a \(\sqrt{a^2 – x^2}\) or \(a^2 + x^2\)? | Inverse trig rules or trig substitution |
| 8 | None of the above? | It may have no formula. Use numbers. |
Question 1: Can You Simplify It First?
Many integrals that look hard are easy after one line of algebra. Expand products, split fractions with a single term on the bottom, and rewrite roots as powers.
Example: \(\int \frac{(x+1)^2}{x}\,dx\). Expand and split: \(\frac{x^2 + 2x + 1}{x} = x + 2 + \frac1x\). Now each piece is basic:
\[\int \left(x + 2 + \frac1x\right)dx = \frac{x^2}{2} + 2x + \ln|x| + C.\]
Question 2: Is Every Piece in the Basic Table?
If the function is a sum of powers, exponentials, \(\frac1x\), sines and cosines, the antiderivative rules finish the job term by term.
Example: \(\int \left(3\sqrt{x} – 2e^x + \cos x\right)dx = 2x^{3/2} – 2e^x + \sin x + C.\)
Question 3: Is There an Inside Function With Its Derivative Nearby?
Look for a function inside another one (inside a power, a root, \(e\), sin or cos, or the bottom of a fraction) whose derivative also appears as a factor. That is the signal for u-substitution.
Example: \(\int \frac{\cos x}{\sin x}\,dx\). The top is the derivative of the bottom. With \(u = \sin x\):
\[\int \frac{du}{u} = \ln|\sin x| + C.\]
Question 4: Is It a Product of Two Different Kinds of Functions?
A polynomial times an exponential, a polynomial times a trig function, or a lone logarithm calls for integration by parts. Choose \(u\) with LIATE.
Example: \(\int x\cos x\,dx\). With \(u = x\) and \(dv = \cos x\,dx\):
\[\int x\cos x\,dx = x\sin x + \cos x + C.\]
Question 5: Is It a Fraction of Two Polynomials?
If the top’s degree is at least the bottom’s, divide first. Then factor the bottom and split into partial fractions.
Example: \(\int \frac{1}{x^2 – 1}\,dx\). Factor and split: \(\frac{1}{(x-1)(x+1)} = \frac{1/2}{x-1} – \frac{1/2}{x+1}\). Then
\[\int \frac{1}{x^2-1}\,dx = \frac12\ln|x-1| – \frac12\ln|x+1| + C.\]
The partial fraction calculator does the splitting step for you.
Question 6: Is It a Power of a Trig Function?
Squares of sine and cosine need the half-angle identities. Odd powers use \(\sin^2 x + \cos^2 x = 1\) and then a substitution.
Example: \(\int \cos^2 x\,dx\). Use \(\cos^2 x = \frac{1 + \cos 2x}{2}\):
\[\int \cos^2 x\,dx = \frac{x}{2} + \frac{\sin 2x}{4} + C.\]
Question 7: Do You See \(a^2 + x^2\) or \(\sqrt{a^2 – x^2}\)?
These shapes lead to inverse trig functions.
Example: \(\int \frac{1}{\sqrt{4 – x^2}}\,dx = \sin^{-1}\frac{x}{2} + C\) and \(\int \frac{1}{4 + x^2}\,dx = \frac12\tan^{-1}\frac{x}{2} + C\).
Harder versions, like \(\int \sqrt{4 – x^2}\,dx\), use a trig substitution such as \(x = 2\sin\theta\).
Question 8: Still Stuck? It Might Have No Formula
Some simple-looking functions, such as \(e^{x^2}\), \(\frac{\sin x}{x}\) and \(\sqrt{1 + x^3}\), have no antiderivative made of ordinary functions. This is a proven fact, not a gap in your skills. For these, a definite integral can still be found numerically. If a good antiderivative calculator cannot find a formula either, this is usually why.
Always Finish With a Check
Whatever method you used, differentiate your answer. If you get the original function back, you are done. If not, the difference usually points straight to the mistake. See how to check an antiderivative for more tricks.
Quiz: Which Method Would You Use?
Decide on the method before you open each answer.
1. \(\int x^2 e^{x^3}\,dx\)
U-substitution with \(u = x^3\): \(\frac13e^{x^3} + C\)
2. \(\int x^2 e^{x}\,dx\)
Integration by parts (twice): \(e^x(x^2 – 2x + 2) + C\)
3. \(\int \frac{x^3 + x}{x}\,dx\)
Simplify first: \(\int (x^2 + 1)\,dx = \frac{x^3}{3} + x + C\)
4. \(\int \frac{3}{x^2 + 9}\,dx\)
Inverse trig rule: \(\tan^{-1}\frac{x}{3} + C\)
5. \(\int \sin^2 x\,dx\)
Half-angle identity: \(\frac{x}{2} – \frac{\sin 2x}{4} + C\)
Calculators for This Topic
Practice what you just read. Each tool shows full steps and checks its own answer.
- ∫Antiderivative Calculator – Find any antiderivative with full steps and a check.
- uU-Substitution Calculator – See how to pick u and du for any substitution.
- ∂Integration by Parts Calculator – LIATE choice, u, dv, v and du shown clearly.
Frequently Asked Questions
How do you find the antiderivative of a function?
Simplify it, then match it to a rule: basic rules for powers, exponentials and trig functions, u-substitution when an inside function and its derivative appear, integration by parts for products, and partial fractions for rational functions. Add \(+C\) and check by differentiating.
What is the general antiderivative?
It is the antiderivative written with \(+C\), such as \(x^3 + C\) for \(3x^2\). It represents every possible antiderivative at once.
How do I find the antiderivative of a fraction?
If the bottom is a single term, split the fraction. If the top is the derivative of the bottom, the answer is \(\ln|\text{bottom}| + C\). Otherwise, factor the bottom and use partial fractions.
What if two methods both seem to work?
Try the faster one first, usually substitution. Both methods give equivalent answers, sometimes differing only by a constant.
Can every function be integrated?
Every continuous function has an antiderivative, but some of them cannot be written with ordinary functions. Examples are \(e^{x^2}\) and \(\frac{\sin x}{x}\).
