<<–2/”>a href=”https://exam.pscnotes.com/5653-2/”>h2>ILATE: A Rule for Integration by Parts
Integration by parts is a powerful technique for evaluating integrals that involve the product of two functions. The ILATE rule is a mnemonic device that helps you choose which function to differentiate and which to integrate when applying integration by parts.
What is ILATE?
ILATE stands for:
- Inverse trigonometric functions
- Logarithmic functions
- Algebraic functions
- Trigonometric functions
- Exponential functions
This order represents the priority for choosing the function to be differentiated (u) in the integration by parts formula:
â« u dv = uv – â« v du
How to Use ILATE
- Identify the two functions: Determine the two functions that are being multiplied in the integrand.
- Apply ILATE: Choose the function that appears earlier in the ILATE list as ‘u’ (the function to be differentiated). The remaining function will be ‘dv’ (the function to be integrated).
- Apply the integration by parts formula: Substitute the chosen ‘u’ and ‘dv’ into the formula and solve the integral.
Examples of ILATE in Action
Example 1: â« x sin(x) dx
- Identify the functions: x (algebraic) and sin(x) (trigonometric)
- Apply ILATE: Algebraic functions come before trigonometric functions in ILATE, so u = x and dv = sin(x) dx.
- Apply the formula:
- du = dx
- v = -cos(x)
- â« x sin(x) dx = x(-cos(x)) – â« -cos(x) dx = -x cos(x) + sin(x) + C
Example 2: â« ln(x) dx
- Identify the functions: ln(x) (logarithmic) and 1 (algebraic)
- Apply ILATE: Logarithmic functions come before algebraic functions in ILATE, so u = ln(x) and dv = 1 dx.
- Apply the formula:
- du = (1/x) dx
- v = x
- â« ln(x) dx = ln(x) * x – â« x * (1/x) dx = x ln(x) – x + C
Example 3: â« e^x cos(x) dx
- Identify the functions: e^x (exponential) and cos(x) (trigonometric)
- Apply ILATE: Exponential functions come after trigonometric functions in ILATE, so u = cos(x) and dv = e^x dx.
Apply the formula:
- du = -sin(x) dx
- v = e^x
- â« e^x cos(x) dx = cos(x) * e^x – â« e^x (-sin(x)) dx = e^x cos(x) + â« e^x sin(x) dx
Notice that the integral on the right-hand side still involves a product of functions. We need to apply integration by parts again, this time with u = sin(x) and dv = e^x dx.
- â« e^x sin(x) dx = sin(x) * e^x – â« e^x cos(x) dx
Substituting this back into the original equation, we get:
- â« e^x cos(x) dx = e^x cos(x) + sin(x) * e^x – â« e^x cos(x) dx
Solving for the original integral:
- 2â« e^x cos(x) dx = e^x cos(x) + sin(x) * e^x
- â« e^x cos(x) dx = (e^x cos(x) + sin(x) * e^x) / 2 + C
When ILATE Doesn’t Work
While ILATE is a useful guideline, it’s not always the best choice. Sometimes, other strategies might be more efficient. For example:
- If the integral is simpler to solve by differentiating the other function: In some cases, differentiating the function that comes later in ILATE might lead to a simpler integral.
- If the integral involves a function that doesn’t fit into ILATE: For example, if the integral involves a function like arctan(x), you might need to use a different approach.
Table 1: ILATE Hierarchy
| Category | Function | Example |
|---|---|---|
| Inverse Trigonometric | arcsin(x), arccos(x), arctan(x) | â« arcsin(x) dx |
| Logarithmic | ln(x), log(x) | â« ln(x) dx |
| Algebraic | x, x^2, âx | â« x^2 sin(x) dx |
| Trigonometric | sin(x), cos(x), tan(x) | â« sin(x) e^x dx |
| Exponential | e^x, a^x | â« e^x cos(x) dx |
Table 2: Integration by Parts Formula
| Formula | Description |
|---|---|
| â« u dv = uv – â« v du | The integration by parts formula, where u and dv are functions of x. |
Frequently Asked Questions (FAQs)
Q: What if I have more than two functions in the integrand?
A: You can apply integration by parts multiple times, choosing a different ‘u’ and ‘dv’ each time.
Q: Can I use ILATE in reverse order?
A: Yes, you can try differentiating the function that comes later in ILATE. However, this might not always lead to a simpler integral.
Q: What if I get stuck after applying ILATE?
A: If you’re unable to solve the integral after applying ILATE, consider trying a different integration technique or using a table of integrals.
Q: Is ILATE the only way to solve integration by parts problems?
A: No, ILATE is just a mnemonic device to help you choose ‘u’ and ‘dv’. You can also use other strategies, such as choosing the function that becomes simpler after differentiation.
Q: What are some other integration techniques?
A: Other integration techniques include:
- Substitution: Replacing a part of the integrand with a new variable.
- Partial FRACTIONS: Decomposing a rational function into simpler fractions.
- Trigonometric Substitution: Using trigonometric identities to simplify the integral.
Q: Where can I find more examples and practice problems?
A: You can find numerous examples and practice problems in calculus textbooks, online Resources, and practice websites.