ILATE Full Form

July 27, 2024 by rawan239

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: ∫ 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)
  • ∫ x sin(x) dx = x(-cos(x)) – ∫ -cos(x) dx = -x cos(x) + sin(x) + C

Example 2: ∫ 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
  • ∫ ln(x) dx = ln(x) * x – ∫ x * (1/x) dx = x ln(x) – x + C

Example 3: ∫ 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
  • ∫ e^x cos(x) dx = cos(x) * e^x – ∫ e^x (-sin(x)) dx = e^x cos(x) + ∫ 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.

  • ∫ e^x sin(x) dx = sin(x) * e^x – ∫ e^x cos(x) dx

  Substituting this back into the original equation, we get:

  • ∫ e^x cos(x) dx = e^x cos(x) + sin(x) * e^x – ∫ e^x cos(x) dx

  Solving for the original integral:

  • 2∫ e^x cos(x) dx = e^x cos(x) + sin(x) * e^x
  • ∫ 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)	∫ arcsin(x) dx
Logarithmic	ln(x), log(x)	∫ ln(x) dx
Algebraic	x, x^2, √x	∫ x^2 sin(x) dx
Trigonometric	sin(x), cos(x), tan(x)	∫ sin(x) e^x dx
Exponential	e^x, a^x	∫ e^x cos(x) dx

Table 2: Integration by Parts Formula

Formula	Description
∫ u dv = uv – ∫ 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.