Integration by Parts Infographic

Integration by parts is a method for evaluating integrals of products of functions. It is especially useful when one factor becomes simpler after differentiation while the other is easy to integrate. This technique comes directly from the product rule for derivatives, so it connects differentiation and integration in a very natural way.

Students use it often in calculus, differential equations, and physics.

The core formula is the $\int u \, dv = uv - \int v \, du$ . You choose one part of the integrand to be $u$ and the other to be $dv$ , then compute $du$ and $v$ . A good choice makes the new integral simpler than the original one.

Common examples include products such as $x e^x$ , $x \sin x$ , and $x \ln x$ , and repeated use can handle higher powers like $x^2 e^x$ .

Understanding Integration by Parts

The LIATE list is useful because different function families behave differently under differentiation and integration. Logarithms usually become simpler when differentiated, since the derivative of natural log of x is one over x. Algebraic factors such as x squared lose a power each time they are differentiated.

In contrast, an exponential often stays essentially unchanged after differentiation, while integrating it is easy. This creates the trade off that makes the method work.

LIATE is only a starting guide. A choice that follows the list but produces a harder remaining integral is not a good choice.

Keep track of the two transformations separately. One chosen part is differentiated to produce du. The other is integrated to produce v.

The new integral contains v multiplied by du, so both results matter. A frequent error is to differentiate or integrate the wrong factor after making the choice. Another common error is losing the subtraction sign.

The product term comes first, then the remaining integral is subtracted. Writing each line clearly helps prevent sign mistakes, especially when trigonometric functions are involved. The integral of sine is negative cosine, so one missed negative can change the whole answer.

Repeated use works because powers of x eventually disappear under differentiation. For an expression containing x squared times an exponential, choosing the polynomial as u reduces x squared to two x, then to a constant. Each round produces a new product term and a smaller integral.

With sine or cosine, the pattern can take several steps because these functions cycle when differentiated or integrated. If the original integral returns after several rounds, move that returned integral to the other side and combine like terms. This is a normal outcome for products involving exponentials with sine or cosine.

Definite integrals need extra care because the product term is evaluated at both endpoints. First form the product of u and v. Then substitute the upper endpoint and subtract the value at the lower endpoint.

Only after that should the remaining definite integral be evaluated. This method appears in physics whenever a changing quantity is multiplied by another changing quantity. It can describe work when force varies with position, or accumulated effects in motion models.

The calculation matters because a product can hide two linked changes. Integration by parts separates those changes into a boundary contribution and a contribution spread across the interval.

When checking an answer, differentiate the final expression. A correct derivative returns the original product, which is the most reliable test.

Key Facts

Integration by parts formula: $\int u \, dv = uv - \int v \, du$
It comes from the product rule: $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$
Definite integral form: $\int_a^b u \, dv = [uv]_a^b - \int_a^b v \, du$
Choose u so that differentiating it simplifies it, and choose dv so that integrating it is straightforward
A common guide is LIATE: Logarithmic, Inverse trig, Algebraic, Trig, Exponential
Repeated integration by parts is often needed for integrals like $\int x^n e^x \, dx$ or $\int x^n \sin x \, dx$

Vocabulary

Integrand: The integrand is the expression inside an integral that is being integrated.
u: In integration by parts, u is the factor chosen to be differentiated into du.
dv: In integration by parts, dv is the factor chosen to be integrated into v.
Product rule: The product rule states that $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$ for two functions multiplied together.
LIATE: LIATE is a guideline for choosing u by preferring logarithmic, inverse trigonometric, algebraic, trigonometric, then exponential functions.

Common Mistakes to Avoid

Choosing $u$ and $dv$ poorly, which makes the new integral harder than the original. Pick $u$ so differentiation simplifies it and $dv$ so integration is easy.
Forgetting the minus sign in the formula, which changes the final answer. Write the full formula before substituting.
Differentiating or integrating the chosen parts incorrectly, especially when finding du or v. Check each derivative and antiderivative separately before continuing.
Ignoring the boundary term in definite integrals, which loses part of the answer. After finding $uv$ , evaluate $[uv]$ from $a$ to $b$ before computing the remaining integral.

Practice Questions

1 Use integration by parts to evaluate the integral of $x e^x \, dx$ .
2 Use integration by parts to evaluate the integral of $x \cos x \, dx$ .
3 For the integral of $x \ln x \, dx$ , explain why choosing $u = \ln x$ and $dv = x \, dx$ is usually better than choosing $u = x$ and $dv = \ln x \, dx$ .

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Understanding Integration by Parts

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