Integration by Parts Calculator

Enter the evaluated uv and ∫ v du values to compute the integration by parts result using ∫ u dv = uv − ∫ v du. Optionally expand symbolic labels to record your u, dv, uv, and v du expressions.

Created by Luis GonzalezLast updated: June 5, 2026

How to Use This Calculator

1
Enter the Evaluated uv
Provide the numerical value of the uv term after substituting bounds or simplifying, e.g., 20.
2
Enter the Evaluated ∫ v du
Provide the numerical value of the ∫ v du integral after evaluation, e.g., 7.5.
3
Optionally expand Symbolic Labels
Click 'Show symbolic labels' to record the u, dv, uv, and v du expressions for reference. These are not used in calculations.
4
Review your results
Review the ∫ u dv (Result), uv Term, and ∫ v du Correction cards. After calculating, use Recent to revisit previous scenarios.

Example Calculation

A calculus student is evaluating ∫ x·e^x dx over specific bounds and has determined that the uv term evaluates to 20 and the ∫ v du term evaluates to 7.5.

Evaluated uv

Evaluated ∫ v du

7.5

Results

∫ u dv (Result)

12.5; uv Term: 20; ∫ v du Correction: 7.5.

Tips

Use LIATE to choose u

Prioritize choosing u using the LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Functions earlier in this list simplify when differentiated, making the remaining ∫ v du easier to solve.

Check your sign

The formula subtracts ∫ v du from uv. If your result is unexpectedly negative, verify that you haven't swapped the two evaluated terms. The uv term goes in the first field, the integral correction in the second.

Record your work with symbolic labels

Expand the symbolic labels section to note your u, dv, uv, and v du expressions alongside the numerical values. This makes it easy to review your work and compare scenarios in the Recent history.

Don't forget the constant of integration

For indefinite integrals, remember to add + C to your final result. The calculator shows the numerical evaluation; the constant accounts for any vertical shift in the antiderivative.

Understanding Integration by Parts

The Integration by Parts Calculator applies the fundamental formula ∫ u dv = uv − ∫ v du to your pre-evaluated numerical terms. Enter the values of the uv product and the ∫ v du correction integral, and the calculator returns the final result along with a breakdown of each component. This technique is a cornerstone of calculus, essential for solving integrals involving products of functions in physics, engineering, and applied mathematics.

The Integration by Parts Formula

Integration by parts is derived from the product rule of differentiation and transforms a difficult integral into a potentially simpler one. You choose which part of the integrand to differentiate (u) and which to integrate (dv), then apply:

∫ u dv = uv − ∫ v du

The calculator evaluates this as:

Result = Evaluated uv − Evaluated ∫ v du

Evaluated uv is the numerical value of u times v (after substituting bounds for definite integrals or simplifying for indefinite ones), and Evaluated ∫ v du is the numerical value of the remaining integral after performing the integration.

💡 For analyzing the behavior of functions and their derivatives, our Curve Sketching Summary Calculator can provide visual insights into mathematical properties.

Worked Example: Evaluating ∫ x·e^x dx

A calculus student needs to evaluate an integral using integration by parts. They have chosen u = x and dv = e^x dx, and after working through the symbolic steps, determined that the uv term evaluates to 20 and the ∫ v du term evaluates to 7.5.

Identify the formula: ∫ u dv = uv − ∫ v du
Substitute evaluated terms: ∫ u dv = 20 − 7.5
Calculate: 20 − 7.5 = 12.5
Interpret: The result is positive (12.5), meaning the uv term (20) dominates the ∫ v du correction (7.5). The uv term is 2.67 times larger than the correction.

The calculator displays three cards: ∫ u dv (Result) = 12.5, uv Term = 20, and ∫ v du Correction = 7.5.

💡 For physical systems that involve integral calculus, our Damped Oscillation Calculator helps analyze the behavior of oscillating systems where integration by parts appears in solving differential equations.

Strategies for Choosing u and dv

The success of integration by parts depends on choosing u and dv wisely. The LIATE mnemonic suggests choosing u from the earliest applicable category:

Logarithmic functions (e.g., ln x)
Inverse trigonometric functions (e.g., arctan x)
Algebraic functions (e.g., x, x^2)
Trigonometric functions (e.g., sin x, cos x)
Exponential functions (e.g., e^x, 2^x)

For ∫ x·e^x dx, x (Algebraic) comes before e^x (Exponential), so u = x and dv = e^x dx. Differentiating u gives du = dx (simpler), and integrating dv gives v = e^x (straightforward). The new integral ∫ e^x dx is much simpler than the original.

When Integration by Parts Must Be Applied Repeatedly

Some integrals require multiple applications of integration by parts, or a technique called tabular integration. For example, ∫ x^2·e^x dx requires two rounds: the first reduces x^2 to x, and the second reduces x to 1. Each round produces a new uv term and a new ∫ v du that becomes the next integral. You can use this calculator at each stage, entering the evaluated terms for that round, and save each step using the Recent history to track your work through the full solution.

Frequently Asked Questions

What is integration by parts used for?

Integration by parts is a calculus technique for integrating products of functions that cannot be simplified by substitution alone. It transforms a complex integral into the product uv minus a potentially simpler integral ∫ v du. Common applications include ∫ x·e^x dx, ∫ x·sin(x) dx, and ∫ ln(x) dx.

What is the LIATE rule?

LIATE is a mnemonic for choosing u in integration by parts: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. The function appearing earliest in this list should typically be u because differentiating it simplifies the expression. For ∫ x·e^x dx, x (Algebraic) comes before e^x (Exponential), so u = x.

Can integration by parts be applied to definite integrals?

Yes. For definite integrals, the uv term is evaluated at the upper and lower bounds (as [uv]_a^b), and ∫ v du is also evaluated over the same limits. Enter the evaluated numerical values of both terms into this calculator to get the definite integral result.

What do the symbolic label fields do?

The symbolic labels (u, dv, uv, v du) are for your reference only — they let you record the expressions you used so you can review your work or compare different setups in the Recent history. They are not parsed or used in the numerical calculation.

Why does the calculator show only three result cards?

The calculator applies the formula ∫ u dv = uv − ∫ v du. The three cards show the final result, the uv term, and the ∫ v du correction. Each card includes a subheader indicating which term dominates and whether the result is positive or negative, giving you a clear breakdown of the computation.