How does integration by parts work?

In summary, integration by parts is a method of integration that is used when the integrand can be written in the form of u(x) * dv/dx. This method involves using the product rule of differentiation in reverse to simplify the integral. The steps involved include choosing u(x) and dv/dx, applying the product rule, and then integrating the resulting equation. If needed, this process may need to be repeated multiple times until the integral can be solved. Other methods, such as substitution, can also be used to solve integrals.
  • #36
help with integration by parts *urgent*

i need help with this because it's driving me crazy.
I need to integrate:

X^3(e^(3x^2))

I can do simpler integration by parts but i can't get this one to work out. The answer works out to:

[(x^2)/6 -1/18]e^(3x^2) please be very detailed, because I know how to do simpler integration by parts, but this example is proving to be very difficult.
 
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  • #37
Put this into a separate post
 
<h2>1. How does integration by parts work?</h2><p>Integration by parts is a technique used in calculus to evaluate integrals of products of functions. It is based on the product rule for differentiation, and involves finding an antiderivative by breaking down the integral into simpler parts.</p><h2>2. What is the formula for integration by parts?</h2><p>The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are functions and dv and du are their differentials.</p><h2>3. When should integration by parts be used?</h2><p>Integration by parts should be used when the integral involves a product of two functions, and one of the functions has a simpler antiderivative than the other. It is also useful when the integral cannot be evaluated by substitution or other techniques.</p><h2>4. How do you choose which function to assign as u and which as dv in integration by parts?</h2><p>The choice of u and dv is based on the LIPET rule, which stands for Logarithmic, Inverse trigonometric, Polynomial, Exponential, and Trigonometric functions. In general, u should be chosen as the function that will simplify after repeated differentiation, and dv should be chosen as the more complicated function.</p><h2>5. What are the common mistakes to avoid when using integration by parts?</h2><p>Some common mistakes to avoid when using integration by parts include forgetting to apply the formula correctly, choosing the wrong function for u, and making mistakes in the algebraic manipulation. It is important to carefully check each step and be familiar with the properties of the functions involved.</p>

1. How does integration by parts work?

Integration by parts is a technique used in calculus to evaluate integrals of products of functions. It is based on the product rule for differentiation, and involves finding an antiderivative by breaking down the integral into simpler parts.

2. What is the formula for integration by parts?

The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are functions and dv and du are their differentials.

3. When should integration by parts be used?

Integration by parts should be used when the integral involves a product of two functions, and one of the functions has a simpler antiderivative than the other. It is also useful when the integral cannot be evaluated by substitution or other techniques.

4. How do you choose which function to assign as u and which as dv in integration by parts?

The choice of u and dv is based on the LIPET rule, which stands for Logarithmic, Inverse trigonometric, Polynomial, Exponential, and Trigonometric functions. In general, u should be chosen as the function that will simplify after repeated differentiation, and dv should be chosen as the more complicated function.

5. What are the common mistakes to avoid when using integration by parts?

Some common mistakes to avoid when using integration by parts include forgetting to apply the formula correctly, choosing the wrong function for u, and making mistakes in the algebraic manipulation. It is important to carefully check each step and be familiar with the properties of the functions involved.

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