u-Substitution or Integration by Parts? How to Choose

Differentiation is a procedure. Integration is a search.

That difference is why this question feels harder than the equivalent one about derivatives. With a derivative you can always look at a function and determine what to do, because every rule has a trigger you can check. With an integral you often have to try something, watch it fail, and go back.

This is not a personal failing, and the standard references say so out loud. Paul's Online Math Notes introduces its integration strategy by warning that "it isn't a hard and fast set of rules for determining the method that should be used. It is really nothing more than a general set of guidelines." OpenStax, on choosing between techniques, allows simply that "sometimes it is a matter of trial and error."

So the flailing is the method, not a symptom. There is still a sensible order to try things in, and one test that resolves most of the u-substitution question outright.

The test that settles substitution

Look for a composite function, something tucked inside something else. Then ask whether the derivative of that inside thing is also sitting in the integrand, give or take a constant. OpenStax states the requirement plainly in its problem-solving strategy for substitution: select an expression to call $u$ "such that $g'(x)$ is also part of the integrand."

Paul's puts the same test the other way round, which some people find easier to run: pretend you were about to differentiate the integrand instead, and ask whether there would be a chain rule. If there would be, the inside function is very likely your substitution.

If it is, substitute. If it is not, substitution will not save you and you can stop considering it.

Take this one:

$$\int 2x\cos(x^2)\,dx$$

The inside here is $x^2$, and its derivative is $2x$. Look at the integrand: there is a $2x$ sitting right there. So set $u = x^2$, which makes $du = 2x\,dx$, and the whole thing collapses:

$$\begin{aligned} \int 2x\cos(x^2)\,dx &= \int \cos u\,du \\ &= \sin u + C \\ &= \sin(x^2) + C \end{aligned}$$

The constant does not matter, only the shape. If the integrand had been $x\cos(x^2)$ instead, you would still substitute and simply carry a factor of one half, giving $\frac{1}{2}\sin(x^2) + C$.

The pair worth memorising

These two look almost identical and take opposite methods, which is the clearest demonstration I know of why appearance is not the signal.

$$\int x\cos(x^2)\,dx \qquad\text{and}\qquad \int x\cos(x)\,dx$$

In the first, $x^2$ is an inside function and the stray $x$ is very nearly its derivative. Substitution, and it falls apart in one line.

In the second, there is no inside function at all. The $x$ and the $\cos(x)$ are two unrelated things multiplied together, and the $x$ is not the derivative of anything nested. Substitution has nothing to grab. This is integration by parts, and it gives $x\sin x + \cos x + C$.

One character different. Completely different method. Run the test rather than trusting the silhouette.

When it is parts

First, the thing students actually want to know. OpenStax answers it in the opening line of its chapter on parts: "Many students want to know whether there is a product rule for integration. There isn't, but there is a technique based on the product rule for differentiation that allows us to exchange one integral for another."

Exchange one integral for another. That is the honest description, and it tells you the whole strategy.

Integration by parts is for a product of two genuinely unrelated functions, where differentiating one of them makes the problem simpler.

That last clause is the whole game. Parts trades your integral for a different integral, so it only helps if the new one is easier. Choosing which factor becomes u is choosing which one you differentiate, and you want to differentiate the thing that improves when differentiated.

Polynomials are the obvious case. In $\int x e^x\,dx$, differentiating $x$ gives 1 and the $x$ disappears. Let $u = x$ and $dv = e^x\,dx$, and you get $xe^x - e^x + C$. The $x$ is gone, and that is why it worked.

LIATE is the usual mnemonic for picking $u$: logarithmic, inverse trig, algebraic, trigonometric, exponential, in that priority order, with whatever comes first on the list becoming your $u$. It dates back to a 1983 note by Herbert Kasube in the American Mathematical Monthly.

Applied to $\int x\ln x\,dx$, LIATE says the logarithm outranks the polynomial, so $u = \ln x$ and $dv = x\,dx$. That gives

$$\int x\ln x\,dx = \frac{1}{2}x^2\ln x - \frac{x^2}{4} + C$$

Had you chosen the other way round, you would have made the problem worse, which is the mnemonic earning its keep.

Note how carefully OpenStax endorses it, though. The acronym "can often help to take some of the guesswork out of our choices" and "serves as an aid in determining an appropriate choice for $u$." Can often help. Serves as an aid. That is a mnemonic, not a theorem, and the hedging is doing real work.

When you need both

Plenty of integrals want a substitution first and parts afterwards, and students often treat this as evidence they have gone wrong.

Take $\int x^3 e^{x^2}\,dx$. Substitution alone will not finish it, but it does simplify it. Let $u = x^2$, so $du = 2x\,dx$ and $x^2$ becomes $u$. Split the $x^3$ into $x^2$ times $x$, and the integral turns into a plain parts problem:

$$\begin{aligned} \int x^3 e^{x^2}\,dx &= \frac{1}{2}\int u e^u\,du \\ &= \frac{1}{2}\left(u e^u - e^u\right) + C \\ &= \frac{1}{2}e^{x^2}\left(x^2 - 1\right) + C \end{aligned}$$

Two techniques, one integral, no error committed. The methods are not rival answers to a question. They are steps you chain.

This particular integral is worth dwelling on, because OpenStax uses it as its own cautionary tale. The example is titled "Applying Integration by Parts When LIATE Doesn't Quite Work," and the explanation is exact: a strict reading of the mnemonic puts A before E, so you would set $u = x^3$ and $dv = e^{x^2}\,dx$. That choice dies immediately, because $\int e^{x^2}\,dx$ cannot be evaluated at all. Their fix is to split the algebraic factor, taking $u = x^2$ and $dv = x e^{x^2}\,dx$, which works and lands on the same answer.

Two things follow. The mnemonic's failure mode is not that it picks a slower route, it is that it can hand you a $dv$ you cannot integrate. And the repair, splitting one factor across $u$ and $dv$, is something no first-letter acronym can express, because LIATE only knows how to rank whole functions.

The same trap shows up in Paul's notes on $\int x^5\sqrt{x^3+1}\,dx$, where taking the whole $x^5$ as $u$ leaves a $dv$ nobody can integrate, and the working choice again splits the powers. Two independent references, one failure mode. When the mnemonic and the integral disagree, the integral is right.

The loop that looks like failure

One case genuinely alarms people. Take $\int e^x\sin x\,dx$, apply parts, and you land on an integral of $e^x\cos x$. Apply parts again to that one, and the thing you originally started with comes straight back at you on the right-hand side.

That looks like you have gone in a circle and wasted five minutes. You have not. Call the original integral $I$, and after the second application you are looking at an equation with $I$ on both sides. Solve it algebraically like any other equation, and you get

$$I = \int e^x\sin x\,dx = \frac{e^x\left(\sin x - \cos x\right)}{2} + C$$

Worth knowing this pattern exists before you meet it under exam conditions, because the natural response to seeing your own question come back at you is to abandon the method that was, in fact, working.

The order to try things

Paul's strategy page diagnoses the most expensive habit students have here, and it is not picking the wrong technique. It is skipping straight to picking one. Most students, he writes, "concentrate almost exclusively" on identifying the type of integral, and "one very large consequence of that exclusion is that often a simple manipulation or substitution is overlooked that could make the integral very easy to do."

His instruction is blunt: "always look for quick, simple substitutions before moving on to the more complicated Calculus II techniques." Given all that, here is the sequence that wastes the least time.

• Can you rewrite it into something standard? Algebra first, always. Expanding a bracket or splitting a fraction finishes more integrals than any technique.
• Is there an inside function whose derivative is present? Substitute.
• Is it a product of unrelated functions where differentiating one simplifies things? Parts, choosing $u$ by LIATE.
• Did substitution simplify without finishing? Follow it with parts.
• Did the original integral reappear? Solve for it algebraically.
• None of the above? A different technique entirely, or no elementary answer exists.

That last point deserves saying plainly, because it is the difference between integration and differentiation and nobody mentions it. Some perfectly ordinary-looking integrals have no answer in elementary functions at all. The integral $\int e^{x^2}\,dx$ is the standard example. You can differentiate anything you can write down. You cannot integrate anything you can write down, and no amount of skill changes that.

Which is the honest reason integration feels different. It is not that you are worse at it. It is a search with dead ends built into the territory.

How to practise the choice

Do what I suggest for derivatives, in reverse. Take twenty integrals and write only the method beside each one, no solving. Substitution, parts, both, or algebra first. Then check.

Selecting the technique is a separate skill from executing it, and it is the one that collapses under time pressure, exactly as reading a function's structure before differentiating is the skill that matters more than knowing the rules. When you want the full working for one, a step-by-step run through an antiderivative will show you whether the method you picked was the fast route or merely a route, and adding bounds changes what the question is asking for without changing which technique gets you there.