Integration by Parts Calculator

Integration by parts is a technique used to find the integral of a product of two functions. It's based on the product rule for differentiation and is particularly useful when the integrand is a product of functions where one is easily differentiable and the other is easily integrable.

What is integration by parts?

Integration by parts is a method for finding antiderivatives based on the product rule for differentiation. The product rule states that if u and v are functions of x, then:

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

Rearranging this equation gives us the integration by parts formula:

∫u dv = uv - ∫v du

This formula allows us to transform an integral of a product into a simpler form involving the product of two functions minus another integral. The choice of u and dv is crucial and should be made based on the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential).

How to use the calculator

Our integration by parts calculator provides a step-by-step solution to help you solve integrals using this technique. Here's how to use it:

Enter the function you want to integrate in the "Function to integrate" field.
Select the variable of integration (usually x).
Choose the lower and upper limits of integration if you want a definite integral.
Click the "Calculate" button to see the step-by-step solution.

The calculator will show you the chosen u and dv, the integration steps, and the final result. You can also visualize the function and its integral with the interactive graph.

Integration by parts formula

The integration by parts formula is derived from the product rule for differentiation. For two functions u(x) and v(x), the formula is:

∫u(x)v'(x) dx = u(x)v(x) - ∫u'(x)v(x) dx

This can also be written as:

∫u dv = uv - ∫v du

Where:

u = first function (choose based on LIATE rule)
dv = differential of the second function
v = integral of dv
du = differential of u

The LIATE rule helps determine which function to choose as u:

Logarithmic functions
Inverse trigonometric functions
Algebraic functions
Trigonometric functions
Exponential functions

Worked example

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

Step 1: Choose u and dv

Using the LIATE rule, we choose u = x (algebraic) and dv = e^x dx (exponential).

Step 2: Find du and v

du = d/dx(x) dx = 1 dx

v = ∫e^x dx = e^x

Step 3: Apply integration by parts formula

∫x e^x dx = x e^x - ∫e^x (1) dx

= x e^x - e^x + C

The final result is x e^x - e^x + C.

Common mistakes

When using integration by parts, there are several common mistakes to avoid:

Choosing u incorrectly: Always use the LIATE rule to select u.
Forgetting to subtract the second integral: The formula requires uv - ∫v du.
Incorrectly differentiating or integrating: Double-check your calculations.
Missing the constant of integration: Always include + C for indefinite integrals.
Applying integration by parts when it's not needed: It's often simpler to use substitution.

Tip: Integration by parts is most effective when one function is easily differentiable and the other is easily integrable. If neither function is easily differentiable or integrable, substitution might be a better approach.

FAQ

When should I use integration by parts?

Use integration by parts when you have a product of functions where one is easily differentiable and the other is easily integrable. It's particularly useful for integrals involving logarithmic, inverse trigonometric, algebraic, trigonometric, or exponential functions.

How do I choose u and dv?

Use the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to select u. Choose the function that comes first in the LIATE order as u. The remaining function becomes dv.

What if I can't find v easily?

If you can't find the integral v easily, you may have chosen u incorrectly. Try choosing a different function as u based on the LIATE rule. If all else fails, substitution might be a better approach.

Can integration by parts be used for definite integrals?

Yes, integration by parts works for both definite and indefinite integrals. For definite integrals, apply the formula and evaluate the antiderivative at the upper and lower limits.

What if I get stuck in a loop?

If you keep getting the same integral after applying integration by parts, you may need to apply it multiple times or consider using a different technique like substitution or partial fractions.