Integration by Parts Calculator

May 9, 2026

Finding an antiderivative of a product like \(x e^x\) or \(\ln x\) often stops students mid‑calculation. Integration by parts transforms these daunting expressions into solvable pieces. Our free Integration by Parts Calculator handles both indefinite and definite integrals while showing every algebraic move.

Common Examples

The calculator accepts standard mathematical notation–polynomials, exponentials, logarithms, trig functions, and their inverses–and automatically selects the optimal split into \(u\) and \(dv\) using the ILATE priority rule. For definite integrals, it substitutes limits after integration and returns a numeric value or simplified expression. Instead of wrestling with repeated substitutions, you see the logic unfold step by step.

What is integration by parts?

Integration by parts is a technique that reverses the product rule for differentiation. The standard formula is:

\[ \int u\,dv = uv - \int v\,du \]

Given an integrand that is a product of two functions, you choose one part as \(u\) (to differentiate) and the other as \(dv\) (to integrate). After computing \(du\) and \(v\), you plug everything into the right‑hand side. The goal is to obtain a new integral \(\int v\,du\) that is simpler than the original.

For definite integrals, the formula extends naturally:

\[ \int*{a}^{b} u\,dv = \Big[uv\Big]*{a}^{b} - \int\_{a}^{b} v\,du \]

Both versions are handled automatically. The method dates back to Brook Taylor (1715) and remains a cornerstone of single‑variable calculus.

When should you use integration by parts?

The method shines when an integrand is a product of two fundamentally different function types, especially when a simple \(u\)-substitution fails. Typical scenarios include:

Products of polynomials and exponentials – \(x e^{2x}\)
Polynomials multiplied by sine or cosine – \(x^2 \cos x\)
Logarithmic functions standing alone – \(\ln x\,dx\) (treat as \(\ln x \cdot 1\,dx\))
Inverse trigonometric functions – \(\arctan x\,dx\)
Products of exponentials and trigonometric functions – \(e^x \sin x\) (cyclic case)

A well‑known mnemonic is the ILATE rule (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential), which ranks candidate functions for \(u\). The calculator follows this order but also permits manual selection when a different split yields a faster solution.

How the calculator handles the integration

You do not need to guess \(u\) and \(dv\) manually. Once you enter the integrand and, optionally, the limits, the engine parses the expression, identifies the function types, and applies ILATE to assign \(u\) and \(dv\). It then differentiates \(u\), integrates \(dv\), and substitutes the pieces into the formula.

For definite integrals, the antiderivative is evaluated at the upper and lower limits. The output displays:

The chosen \(u\) and \(dv\)
The intermediate results \(du\) and \(v\)
The transformed integral \(\int v\,du\) (often simplified)
The final closed‑form answer or numerical value

Cyclic cases (e.g., \(\int e^x \sin x\,dx\)) are detected automatically–the calculator solves the resulting equation for the original integral without looping infinitely.

Step‑by‑step example

Consider \(\int x \cos x\,dx\). The calculator would proceed as follows:

Identify \(u\) and \(dv\) using ILATE: algebraic \(x\) takes priority over trigonometric \(\cos x\), so \(u = x\) and \(dv = \cos x\,dx\).
Differentiate and integrate: \(du = 1\cdot dx\), \(v = \sin x\).
Substitute into the formula: \(\int x \cos x\,dx = x \sin x - \int \sin x\,dx\).
Solve the remaining integral: \(\int \sin x\,dx = -\cos x + C\).
Final answer: \(x \sin x + \cos x + C\).

Working through such an example with the calculator side‑by‑side reinforces the technique far better than passive reading.

Tips for mastering integration by parts

Apply ILATE consistently – Until you develop intuition, the rule prevents dead ends.
Treat “1” as a hidden factor – Logarithms, inverse trigs, and even standalone arctan are really products with 1. Choose \(dv = 1\,dx\) in those cases.
Watch for cyclic patterns – If after two applications the original integral reappears, don’t start over. Add the recurring integral to both sides and solve algebraically.
Use tabular integration for repeated polynomials – When differentiating a polynomial eventually yields zero, tabular integration (the “DI” method) speeds up the work; the calculator employs this shortcut automatically.
Practice with definite integrals – Plugging in limits often reveals cancellations that simplify the arithmetic.

The Integration by Parts Calculator does more than compute–it demonstrates why each step works. Combine tool‑assisted checking with manual practice to turn a challenging technique into a reliable skill.

Frequently Asked Questions

What is the formula for integration by parts?

The core formula is ∫u dv = uv − ∫v du, derived from the product rule for differentiation. You select u and dv from the original integrand, differentiate u to get du, and integrate dv to obtain v. Substituting these pieces into the formula reduces the problem to a (hopefully) simpler integral.

How do I choose u and dv in integration by parts?

Apply the ILATE priority order: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. The function type higher on the list usually becomes u, while the remaining factor becomes dv. The calculator follows this rule automatically, but manual overrides are also supported when needed.

Can integration by parts solve definite integrals?

Yes, the definite integral version is ∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du. After finding the antiderivative, the calculator evaluates it at the upper and lower limits. The step-by-step output shows both the antiderivative and the evaluated numerical result.

What if integration by parts leads to the original integral?

Cyclic cases like ∫eˣ sin x dx require applying the method twice. The resulting equation contains the original integral on both sides. The calculator detects such loops, rearranges algebraically, and solves for the integral directly without infinite repetition.

Is the calculator free to use?

Completely free, no registration or downloads. It runs in the browser and handles unlimited calculations. All steps are displayed transparently, making it suitable for both quick answers and learning the technique.

What types of functions can I integrate?

The tool handles polynomials, exponentials, logarithms, trigonometric, inverse trigonometric functions, and their combinations. Typical inputs include x²eˣ, ln x, arctan x, or x sin x. Hyperbolic functions are also supported in the 2026 version.

Does the calculator show all steps?

Yes. It identifies u and dv, computes du and v, substitutes into the formula, and simplifies. For definite integrals, it additionally substitutes limits and calculates the numeric value. Every algebraic manipulation is displayed.