Integral by Parts Calculator

Integral by parts is a technique used to find the integral of a product of two functions. This calculator helps you solve integrals of the form ∫u dv = uv - ∫v du, where u and dv are functions you choose based on the integral you're solving.

What is Integral by Parts?

Integration by parts is a method for finding the integral of a product of two functions. It's based on the product rule for differentiation and is particularly useful when dealing with products of polynomials, trigonometric functions, and exponential functions.

The formula for integration by parts is:

∫u dv = uv - ∫v du

Where:

u is a function that becomes simpler when differentiated
dv is a function that becomes simpler when integrated

This technique is often used when the integral cannot be found using basic integration rules or substitution.

How to Use the Calculator

Our integral by parts calculator makes solving integrals easy. Here's how to use it:

Enter the function you want to integrate in the "Function" field
Select the appropriate integration variable (usually x)
Choose the functions u and dv that will work best for your integral
Click "Calculate" to see the result
Review the step-by-step solution and the final answer

The calculator will show you the intermediate steps and the final result, helping you understand how the integration by parts method works.

Formula and Explanation

The integration by parts formula is derived from the product rule for differentiation:

d/dx (uv) = u'v + uv'

Rearranging this gives us the integration by parts formula:

∫u dv = uv - ∫v du

To use this formula effectively:

Choose u such that its derivative u' is simpler than u
Choose dv such that its integral v is simpler than dv
Apply the formula and solve the resulting integral

This method is particularly useful for integrals involving products of polynomials, logarithmic functions, and trigonometric functions.

Worked Example

Let's solve the integral ∫x e^x dx using integration by parts.

We'll choose:

u = x (since its derivative is simpler)
dv = e^x dx (since its integral is simpler)

Then:

du = dx
v = e^x

Applying the integration by parts formula:

∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C

So the final answer is x e^x - e^x + C.

This example shows how integration by parts can simplify complex integrals into more manageable forms.

Common Mistakes

When using integration by parts, it's easy to make some common mistakes:

Choosing u and dv incorrectly - they should be chosen to simplify the integral
Forgetting to include the constant of integration C
Making errors in differentiation or integration steps
Applying the formula incorrectly - remember it's uv - ∫v du, not uv + ∫v du

Our calculator helps avoid these mistakes by showing each step clearly and providing the correct final answer.

FAQ

When should I use integration by parts?

Integration by parts is particularly useful when dealing with products of functions, especially when the integral cannot be found using basic integration rules or substitution. It's commonly used with polynomials, trigonometric functions, and exponential functions.

How do I choose u and dv?

Choose u to be a function that becomes simpler when differentiated, and dv to be a function that becomes simpler when integrated. Common choices include polynomials, logarithmic functions, and trigonometric functions.

What if the integral doesn't simplify?

If the integral doesn't simplify after applying integration by parts, you may need to try a different choice of u and dv or consider using another integration technique like substitution or partial fractions.

Can integration by parts be used for definite integrals?

Yes, integration by parts can be used for definite integrals. The formula remains the same, but you'll need to evaluate the antiderivative at the upper and lower limits.