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Integration by Parts (LIATE) Reference cheat sheet - grade 11-12

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Calculus Grade 11-12

Integration by Parts (LIATE) Reference Cheat Sheet

A printable reference covering integration by parts, LIATE, repeated parts, tabular method, and common antiderivative patterns for grades 11-12.

TopicsIntegralsDerivativesFunctionsProcessStudy GuideCalculus
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Integration by parts is a method for finding antiderivatives of products, especially when substitution does not simplify the integral. This cheat sheet helps students choose which factor should be uu and which should be dvdv using the LIATE guideline. It also organizes repeated integration by parts and the tabular method so multi-step problems are easier to manage.

Students need these tools for polynomial, exponential, logarithmic, inverse trig, and trigonometric products.

The main formula comes from the product rule and is written as ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du. LIATE suggests choosing uu in this order: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential. Repeated parts is useful when each step makes uu simpler, such as in ∫x2ex dx\int x^2 e^x\,dx.

The tabular method is a shortcut for repeated parts when derivatives eventually reach 00 or form a simple pattern.

Key Facts

  • The integration by parts formula is ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du.
  • The definite integral form is ∫abu dv=[uv]ab−∫abv du\int_a^b u\,dv = \left[uv\right]_a^b - \int_a^b v\,du.
  • LIATE ranks choices for uu as logarithmic, inverse trigonometric, algebraic, trigonometric, then exponential.
  • After choosing uu, compute dudu by differentiating and compute vv by integrating dvdv.
  • Choose uu so that dudu is simpler than uu, such as choosing u=x3u = x^3 because du=3x2 dxdu = 3x^2\,dx.
  • For ∫xex dx\int x e^x\,dx, choose u=xu = x and dv=ex dxdv = e^x\,dx to get ∫xex dx=xex−ex+C\int x e^x\,dx = x e^x - e^x + C.
  • For ∫ln⁡x dx\int \ln x\,dx, use u=ln⁡xu = \ln x and dv=dxdv = dx to get ∫ln⁡x dx=xln⁡x−x+C\int \ln x\,dx = x\ln x - x + C.
  • In tabular integration, multiply diagonally with alternating signs ++, −-, ++, −- until the derivative column ends or repeats.

Vocabulary

Integration by Parts
A technique for integrating products that rewrites ∫u dv\int u\,dv as uv−∫v duuv - \int v\,du.
LIATE
A guideline for choosing uu in integration by parts, ordered as logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential.
Tabular Method
A shortcut for repeated integration by parts that organizes derivatives, integrals, and alternating signs in a table.
Repeated Integration by Parts
Using integration by parts more than once in the same problem, usually because the remaining integral is still a product.
Differential
A notation such as dudu or dvdv that represents the derivative part used to track substitution and integration steps.
Antiderivative
A function F(x)F(x) whose derivative is the original function, so F′(x)=f(x)F'(x) = f(x).

Common Mistakes to Avoid

  • Choosing uu only because it appears first is wrong because the best uu should usually become simpler when differentiated.
  • Forgetting to integrate dvdv is wrong because the formula requires vv, not just dvdv, in the product uvuv.
  • Dropping the minus sign in uv−∫v duuv - \int v\,du is wrong because it changes the value of the antiderivative.
  • Using LIATE as an absolute rule is wrong because it is a guideline, and some integrals require a different choice to become simpler.
  • Forgetting the constant CC in an indefinite integral is wrong because all antiderivatives differ by a constant.

Practice Questions

  1. 1 Use integration by parts to evaluate ∫xcos⁡x dx\int x\cos x\,dx.
  2. 2 Use LIATE to evaluate ∫x2ex dx\int x^2 e^x\,dx.
  3. 3 Evaluate the definite integral ∫01xe2x dx\int_0^1 x e^{2x}\,dx.
  4. 4 Explain why LIATE usually suggests choosing u=ln⁡xu = \ln x instead of u=xu = x in ∫xln⁡x dx\int x\ln x\,dx.

Understanding Integration by Parts (LIATE) Reference

The method works because differentiation of a product creates two terms. If one factor is difficult to integrate but easy to differentiate, moving the derivative onto that factor can improve the problem. The price is that the other factor must be integrated first.

This trade is the whole idea. A good choice reduces complexity after one step. For example, powers of x shrink when differentiated.

Logarithms become one divided by x. Inverse trigonometric functions often become rational expressions that may fit known antiderivatives. By contrast, differentiating an exponential usually leaves it almost unchanged, so it is rarely the helpful factor to differentiate when another choice exists.

LIATE is a useful first guess, not an unbreakable law. Students should test whether the resulting remaining integral is actually easier. A trigonometric factor may be better as the part to differentiate when its derivative creates a familiar partner.

For products involving sine and cosine, substitution or identities can sometimes be shorter. For a product of two functions that both repeat under differentiation and integration, such as an exponential times sine, integration by parts may return to the original integral. This is not a failure.

Collect the returned integral on one side, then divide to solve for it. These cyclic problems require careful algebra because one missed negative sign changes the answer.

The tabular method is mainly an organized record of repeated work. Its derivative column should contain a function that eventually becomes zero, such as a polynomial. Its integral column should contain a function that remains manageable after several integrations, such as an exponential or sine and cosine.

The alternating signs come from repeatedly subtracting the next leftover integral. They are not a decoration. A reliable habit is to write the signs before making diagonal products.

Stop only when the derivative column reaches zero. If the derivatives repeat instead, the table alone does not finish the problem. The repeated expression must be handled by the collect-and-solve approach.

Definite integrals need one extra layer of attention. Evaluate the product term at the upper and lower limits before subtracting the remaining definite integral. Do not attach an arbitrary constant to a definite-integral answer.

Check that every function is defined across the full interval. For instance, a logarithm requires positive inputs in the usual real-number setting. Integration by parts appears in physics when position, velocity, or force changes with time and a product must be accumulated.

It appears in probability when finding averages from weighted functions. The strongest learning habit is to differentiate the final antiderivative. If its derivative does not recreate the original integrand exactly, revisit the choices of derivative, antiderivative, sign, and constant.