Integration by Parts

LIATE Rule and the IBP Formula

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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 $x^2 e^x$ .

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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