1. What is Integration by Parts?

Integration by parts is a technique used to evaluate integrals where the integrand is a product of two functions. It is derived from the product rule for differentiation and is particularly useful when one function is easily differentiable and the other is easily integrable.

2. How Does the Calculator Work?

The calculator uses the integration by parts formula:

Where:

• \( u \) — The function to differentiate
• \( dv \) — The differential to integrate
• \( du \) — The derivative of u
• \( v \) — The integral of dv

Explanation: The formula transforms the original integral into a potentially simpler integral plus an evaluated term.

3. Importance of Integration by Parts

Details: This method is essential for solving integrals of products of functions, especially when substitution methods fail. It's widely used in calculus, physics, and engineering problems.

4. Using the Calculator

Tips: Enter the function for u (the part you want to differentiate) and the differential for dv (the part you want to integrate). Choose u and dv carefully to simplify the resulting integral.

5. Frequently Asked Questions (FAQ)

Q1: How do I choose u and dv?
A: Follow the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) - choose u as the function that comes first in this order.

Q2: Can integration by parts be used multiple times?
A: Yes, sometimes you need to apply the method repeatedly until you reach a solvable integral.

Q3: What if the integral becomes more complicated?
A: You might need to reconsider your choice of u and dv or combine this method with other techniques like substitution.

Q4: Are there special cases for integration by parts?
A: Yes, for example when integrating functions like e^x*sin(x), you may need to solve for the original integral algebraically.

Q5: When should I not use integration by parts?
A: When simpler methods like u-substitution can be applied, or when the integrand is a simple standard form.