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Integration by Parts Calculator

Verified Calculation Engine

The online Integration by Parts Calculator helps you calculate instantly and solve problems related to Integration. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.

Applies integration by parts: ∫u dv = uv - ∫v du. Choose u and dv using LIATE rule.

Integration by parts: ∫u dv = uv - ∫v du LIATE rule for choosing u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential
Integration by parts for product of functions

Inputs

Please enter a valid Function f(x).
Function to integrate (product of two functions)
First function u (if not provided, will suggest using LIATE)
differential dv (if not provided, will suggest)

Results

Worked Examples
Example 1: Basic

Find ∫x·e^x dx

Inputs:
  • function: x*e^x
Expected Outputs:
  • integral: xe^x - e^x + C = e^x(x - 1) + C
  • uChoice: u = x, dv = e^x dx
  • steps: u = x, dv = e^x dx, so du = dx, v = e^x, then ∫x·e^x dx = xe^x - ∫e^x dx
Example 2: Board Level

Find ∫x·ln(x) dx

Inputs:
  • function: x*ln(x)
Expected Outputs:
  • integral: x²ln(x)/2 - x²/4 + C
  • uChoice: u = ln(x), dv = x dx
  • steps: u = ln(x), dv = x dx, so du = 1/x dx, v = x²/2

About this calculator

Overview

Integration by Parts Calculator evaluates Integration by Parts for the math / integration library. Primary input cue: Function f(x) [required]: Function to integrate (product of two functions). Method anchor: Integration by parts: ∫u dv = uv - ∫v du LIATE rule for choosing u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Integration by pa.

When to use

Use this page when you need integration by parts from known inputs and want a transparent arithmetic or symbolic check.

Inputs explained

  • Function f(x) [required]: Function to integrate (product of two functions)
  • u (Optional) [optional]: First function u (if not provided, will suggest using LIATE)
  • dv (Optional) [optional]: differential dv (if not provided, will suggest)

Formula / method

Integration by parts: ∫u dv = uv - ∫v du LIATE rule for choosing u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Integration by parts for product of functions (calculator id: integration-by-parts-calculator).

Worked example

Worked values: function = x*e^x → integral = xe^x - e^x + C = e^x(x - 1) + C, uChoice = u = x, dv = e^x dx, steps = u = x, dv = e^x dx, so du = dx, v = e^x, then ∫x·e^x dx = xe^x - ∫e^x dx

Interpreting results

Cross-check with a simple numeric example before relying on extreme inputs for integration by parts.

Assumptions

  • Empty required inputs are not silently replaced with zeros for integration by parts. [integration-by-parts-calculator]
  • Integration by Parts is computed only from the fields you enter on this calculator.
  • Real-number arithmetic is used unless integration by parts explicitly needs integers.

Limitations

  • If multiple conventions exist for integration by parts, this calculator uses the convention implemented in its engine path.
  • This page does not provide a full textbook proof for every identity related to integration by parts.
  • Complex/branch cuts and undefined points are not always interactively annotated for integration by parts.

Important note

Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.

How to Use This Calculator

  1. Enter the required values in the input fields.
  2. Click the Calculate button.
  3. View the computed result instantly.

Formula Used

Integration by parts: ∫u dv = uv - ∫v du LIATE rule for choosing u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Integration by parts for product of functions (calculator id: integration-by-parts-calculator).

Example Calculation

Worked values: function = x*e^x → integral = xe^x - e^x + C = e^x(x - 1) + C, uChoice = u = x, dv = e^x dx, steps = u = x, dv = e^x dx, so du = dx, v = e^x, then ∫x·e^x dx = xe^x - ∫e^x dx

Frequently Asked Questions

What is Integration by Parts Calculator?

Integration by Parts Calculator evaluates Integration by Parts for the math / integration library. Primary input cue: Function f(x) [required]: Function to integrate (product of two functions). Method anchor: Integration by parts: ∫u dv = uv - ∫v du LIATE rule for choosing u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Integration by pa.

How does Integration by Parts Calculator work?

Integration by parts: ∫u dv = uv - ∫v du LIATE rule for choosing u: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. Integration by parts for product of functions (calculator id: integration-by-parts-calculator).

Why use this Math Numbers calculator?

Use this page when you need integration by parts from known inputs and want a transparent arithmetic or symbolic check.