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Choosing u and dv for Integration by Parts
Topic 2/3
Choosing $u$ and $dv$ for Integration by Parts
Introduction
Integration by Parts is a fundamental technique in Calculus BC, particularly for solving integrals involving products of functions. Choosing the appropriate $u$ and $dv$ is crucial for simplifying the integral and obtaining a solution efficiently.
Key Concepts
Understanding Integration by Parts
Integration by Parts is derived from the product rule for differentiation. The formula is given by:
$$\int u \, dv = uv - \int v \, du$$
This formula allows for the integration of products of functions by identifying parts of the integrand as $u$ and $dv$, differentiating $u$ to find $du$, and integrating $dv$ to find $v$.
Choosing $u$ and $dv$
Choosing $u$ and $dv$ correctly is essential for simplifying the integral. A common mnemonic to aid in this selection is the LIATE rule, which prioritizes functions based on the category in which they fall.
- L - Logarithmic functions: Functions like $\ln(x)$, $\log(x)$
- I - Inverse trigonometric functions: Functions like $\arctan(x)$, $\arcsin(x)$
- A - Algebraic functions: Polynomials like $x^n$, where $n$ is a real number
- T - Trigonometric functions: Functions like $\sin(x)$, $\cos(x)$
- E - Exponential functions: Functions like $e^x$, $a^x$, where $a$ is a constant
According to LIATE, select $u$ from the function that appears first in the list. For example, given the integral $\int x e^x dx$, $u$ would be $x$ (an algebraic function) and $dv$ would be $e^x dx$ (an exponential function).
Procedure for Integration by Parts
- Identify $u$ and $dv$: Use the LIATE rule or other strategies to select $u$ and $dv$ such that differentiating $u$ and integrating $dv$ simplifies the integral.
- Differentiate and Integrate: Compute $du$ by differentiating $u$, and compute $v$ by integrating $dv$.
- Apply the Formula: Substitute $u$, $dv$, $v$, and $du$ into the Integration by Parts formula.
- Simplify and Integrate: Simplify the resulting integral. If necessary, apply Integration by Parts again.
Examples
Example 1: Compute $\int x e^x dx$.
Solution:
- Choose $u$ and $dv$: Let $u = x$ (algebraic), $dv = e^x dx$ (exponential).
- Differentiate and Integrate: $du = dx$, $v = e^x$.
- Apply the Formula: $$\int x e^x dx = x e^x - \int e^x dx$$ $$= x e^x - e^x + C$$ $$= e^x (x - 1) + C$$
Example 2: Compute $\int \ln(x) dx$.
Solution:
- Choose $u$ and $dv$: Let $u = \ln(x)$ (logarithmic), $dv = dx$ (algebraic).
- Differentiate and Integrate: $du = \frac{1}{x} dx$, $v = x$.
- Apply the Formula: $$\int \ln(x) dx = x \ln(x) - \int x \cdot \frac{1}{x} dx$$ $$= x \ln(x) - \int 1 dx$$ $$= x \ln(x) - x + C$$
Reduction of Integrals
Sometimes, applying Integration by Parts repeatedly can reduce an integral to a more manageable form. This is particularly useful for integrals that do not simplify after one application of the technique.
Tabular Integration by Parts
Tabular Integration is a streamlined method of applying Integration by Parts multiple times. It is especially effective when integrating the product of a polynomial and an exponential or trigonometric function.
- Create a table with derivatives of $u$ and integrals of $dv$.
- Alternate signs and multiply diagonally to sum the terms.
- Continue until the derivatives of $u$ are zero.
Example: Compute $\int x^2 e^x dx$ using Tabular Integration.
Solution:
| Derivatives of $u$ | Integrals of $dv$ |
| $x^2$ | $e^x$ |
| $2x$ | $e^x$ |
| $2$ | $e^x$ |
| $0$ |
Now, apply the alternating signs:
$$
\int x^2 e^x dx = x^2 e^x - 2x e^x + 2 e^x + C = e^x (x^2 - 2x + 2) + C
$$
Comparison Table
| Aspect | Choosing $u$ | Choosing $dv$ |
| Definition | The function to be differentiated to obtain $du$ | The function to be integrated to obtain $v$ |
| Selection Priority | Higher in LIATE | Lower in LIATE |
| Purpose | To simplify the integrand upon differentiation | To provide an integrable component upon integration |
| Pros | Reduces complexity by differentiation | Ensures $v$ is easily integrable |
| Cons | Incorrect choice can complicate the integral | If $dv$ is not easily integrable, the method fails |
Summary and Key Takeaways
- Choosing appropriate $u$ and $dv$ is critical for effective Integration by Parts.
- The LIATE rule serves as a helpful guideline for selecting $u$ and $dv$.
- Proper application simplifies complex integrals, making them manageable.
- Tabular Integration can streamline the process, especially for higher-order polynomials.
- Practice is essential to master the selection and application of Integration by Parts.
Coming Soon!
Examiner Tip
Tips
Always start with the LIATE rule to decide on $u$ and $dv$. Practice with various function types to become familiar with different scenarios. Additionally, check your work by differentiating your final answer to ensure it matches the original integrand, which is especially helpful for AP exam preparation.
Did You Know
Did You Know
Integration by Parts not only aids in solving mathematical integrals but also plays a crucial role in deriving important formulas in physics, such as the Fourier transform. Additionally, the concept extends to infinite-dimensional spaces, impacting areas like quantum mechanics and signal processing.
Common Mistakes
Common Mistakes
One common error is choosing $dv$ incorrectly, leading to more complex integrals. For example, selecting $dv = \sin(x) dx$ instead of $dv = e^x dx$ can complicate the computation. Another mistake is neglecting to simplify the resulting integral after applying the formula, which can make the problem unnecessarily difficult.
FAQ
What is the LIATE rule?
The LIATE rule is a mnemonic that helps prioritize the selection of $u$ in Integration by Parts: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.
When should I use Integration by Parts?
Use Integration by Parts when the integrand is a product of two functions where differentiating one and integrating the other simplifies the integral.
Can Integration by Parts be used more than once on the same integral?
Yes, some integrals require multiple applications of Integration by Parts to reach a solvable form.
What is Tabular Integration?
Tabular Integration is a method that organizes the differentiation and integration steps in a table format, making it easier to apply Integration by Parts multiple times efficiently.
How do I verify my Integration by Parts solution?
Differentiate your final answer to see if you retrieve the original integrand. This step ensures the correctness of your solution.
1.
Integration and Accumulation of Change
2.
Infinite Sequences and Series
3.
Differential Equations
4.
Parametric Equations, Polar Coordinates and Vector-Valued Functions
5.
Applications of Integration
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