Integration by Parts Calculator Step by Step

Integration by parts is a fundamental technique in calculus used to find integrals of products of functions. This method is particularly useful when dealing with functions that are products of polynomials, exponential functions, trigonometric functions, and their combinations.

What is Integration by Parts?

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

The method is derived from the product rule of differentiation, which states that if u and v are functions of x, then:

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

Integrating both sides with respect to x gives:

u v = ∫u' v dx + ∫u v' dx

Rearranging this equation gives the integration by parts formula:

∫u v' dx = u v - ∫u' v dx

Integration by Parts Formula

The integration by parts formula is given by:

∫u dv = u v - ∫v du

Where:

u is a differentiable function of x
dv is the differential of another function v
du is the differential of u
v is an antiderivative of dv

The formula can be remembered using the mnemonic "LIATE" (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to determine which function to choose as u.

How to Use the Calculator

Our integration by parts calculator provides a step-by-step solution for integrals of the form ∫u dv. To use the calculator:

Enter the function u in the first input field
Enter the differential dv in the second input field
Click the "Calculate" button to see the step-by-step solution
Review the result and the detailed steps

The calculator will display the integral, the chosen u and dv, the calculated du and v, and the final result.

Step-by-Step Guide

Step 1: Identify u and dv

Choose u and dv according to the LIATE rule. For example, if the integral is ∫x e^x dx, choose u = x and dv = e^x dx.

Step 2: Differentiate u and Integrate dv

Find du by differentiating u and v by integrating dv. For the example above, du = dx and v = e^x.

Step 3: Apply the Integration by Parts Formula

Substitute u, du, v, and dv into the integration by parts formula: ∫x e^x dx = x e^x - ∫e^x dx.

Step 4: Solve the Remaining Integral

Solve the remaining integral ∫e^x dx to get e^x + C, where C is the constant of integration.

Step 5: Combine Results

Combine the results to get the final integral: x e^x - e^x + C.

Common Examples

Here are some common integrals solved using integration by parts:

Integral	Solution
∫x e^x dx	x e^x - e^x + C
∫x cos x dx	x sin x + cos x + C
∫x^2 e^x dx	(x^2 - 2x + 2) e^x + C
∫ln x dx	x ln x - x + C

FAQ

What is the LIATE rule?: The LIATE rule is a mnemonic used to determine which function to choose as u in integration by parts. It stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.
When should I use integration by parts?: Use integration by parts when dealing with integrals of products of functions, especially when the functions are polynomials, exponential functions, or trigonometric functions.
Can integration by parts be used for all integrals?: No, integration by parts is not a universal method and may not work for all integrals. It is most effective for integrals of products of functions.
What if I choose the wrong u and dv?: If you choose the wrong u and dv, the integral may become more complicated. It's important to choose u and dv according to the LIATE rule to simplify the integral.
How do I know when to stop applying integration by parts?: You should stop applying integration by parts when the remaining integral is simpler and can be solved using basic integration techniques.