Integration by Parts Calculator Step by Step

Integration by parts is a fundamental technique in calculus used to find the integral of a product of two functions. This method is particularly useful when direct integration is difficult or impossible. Our step-by-step guide and interactive calculator will help you master this technique.

What is Integration by Parts?

Integration by parts is a method in calculus that relates the integral of a product of two functions to the product of their antiderivatives. It's based on the product rule for differentiation and is particularly useful for integrals of products of polynomials and transcendental functions.

The method is derived from the product rule for differentiation:

d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Integrating both sides with respect to x gives:

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

Rearranging this equation gives the integration by parts formula:

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

How to Use the Calculator

Our integration by parts calculator makes solving these problems quick and easy. Simply follow these steps:

Enter the function you want to integrate in the "Function to Integrate" field
Select the appropriate parts (u and dv) from the dropdown menus
Click "Calculate" to see the step-by-step solution
Review the result and the detailed explanation

The calculator will show you each step of the integration process, helping you understand how the solution is derived.

Integration by Parts Formula

The general formula for integration by parts is:

∫u dv = uv - ∫v du

Where:

u is a function chosen to differentiate
dv is the differential of another function
v is the antiderivative of dv
du is the differential of u

This formula allows us to transform a difficult integral into one that's easier to solve.

Step-by-Step Guide

Step 1: Identify u and dv

Choose u and dv such that the integral of v du is easier to evaluate than the original integral. Common choices include:

Let u be a polynomial and dv be a transcendental function (like e^x, sin x, etc.)
Let u be a logarithmic function and dv be a polynomial

Step 2: Differentiate and Integrate

Find du by differentiating u and find v by integrating dv.

Step 3: Apply the Formula

Substitute u, dv, du, and v into the integration by parts formula:

∫u dv = uv - ∫v du

Step 4: Evaluate the New Integral

Evaluate the integral ∫v du, which should be easier than the original integral.

Step 5: Combine Results

Combine the results to get the final value of the integral.

Tip: If the new integral ∫v du is still difficult, you may need to apply integration by parts again or use another technique.

Common Mistakes to Avoid

When using integration by parts, be careful about these common errors:

Choosing u and dv incorrectly, making the new integral more difficult than the original
Forgetting to multiply by -1 when rearranging the formula
Making sign errors when differentiating or integrating
Not checking if the integral can be solved by other methods first

Our calculator helps avoid these mistakes by showing each step clearly and allowing you to verify your work.

FAQ

When should I use integration by parts?

Use integration by parts when you have a product of functions to integrate and direct integration is difficult. It's particularly useful for integrals involving polynomials and transcendental 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 can be easily integrated. Common choices are polynomials for u and transcendental functions for dv.

What if the new integral is still difficult?

If the new integral ∫v du is still difficult, you may need to apply integration by parts again or use another technique like substitution or partial fractions.

Can integration by parts be used for definite integrals?

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