MathSimplify
Calculus

Integration by Parts

Solve complex integrals using the integration by parts formula. Get instant step-by-step solutions — completely free, no signup required.

Free Integration by Parts Calculator

Enter your problem below to get an instant step-by-step solution.

About Integration by Parts

Solve complex integrals using the integration by parts formula. Calculus is the mathematical study of continuous change. It provides tools for analyzing functions, computing areas and volumes, and solving differential equations that model real-world phenomena.

Our Integration by Parts calculator handles every type of problem in this topic. Whether you are a student working through homework, preparing for an exam, or a professional needing a quick calculation, you can enter any integration by parts expression and receive a complete, verified solution with every step shown.

Integration by Parts Formula

∫ u dv = uv − ∫ v du

Integration by parts is the product rule run backwards. It is used when the integrand is a product of two different kinds of function, such as x·eˣ or x·ln x. Choose u using the LIATE order (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) so that ∫ v du is simpler than the original.

Worked Example

Evaluate ∫ x·eˣ dx

  1. Let u = x, so du = dx
  2. Let dv = eˣ dx, so v = eˣ
  3. Apply the formula: x·eˣ − ∫ eˣ dx
  4. Integrate the remaining term: x·eˣ − eˣ + C

Answer: eˣ(x − 1) + C

Common Mistakes to Avoid

  • Choosing u = eˣ, which makes the new integral harder rather than easier.
  • Dropping the minus sign in front of ∫ v du.
  • Stopping after one round when the method needs to be applied twice (e.g. ∫ x² eˣ dx).

How to Use the Integration by Parts Calculator

  1. 1

    Enter your calculus expression (e.g., a function for Integration by Parts) into the input field

  2. 2

    Press Enter or click the Solve button to run the calculation

  3. 3

    Review the step-by-step solution that appears below the input

  4. 4

    Check the breakdown of each operation to understand the method

  5. 5

    Try a different problem to practice and build your understanding

Frequently Asked Questions

What is the Integration by Parts formula?

∫ u dv = uv − ∫ v du. Integration by parts is the product rule run backwards. It is used when the integrand is a product of two different kinds of function, such as x·eˣ or x·ln x. Choose u using the LIATE order (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) so that ∫ v du is simpler than the original.

Can you show a worked Integration by Parts example?

Evaluate ∫ x·eˣ dx: Let u = x, so du = dx; Let dv = eˣ dx, so v = eˣ; Apply the formula: x·eˣ − ∫ eˣ dx; Integrate the remaining term: x·eˣ − eˣ + C. Answer: eˣ(x − 1) + C.

Is this Integration by Parts calculator free to use?

Yes — the Integration by Parts calculator on MathSimplify is completely free. There is no signup, no subscription, and no usage limit. Enter any problem and get the step-by-step solution instantly.

Can I use this for Integration by Parts homework and exam practice?

Absolutely. The step-by-step breakdown is designed specifically for learning. You can use it to check your work, understand where you went wrong, or study the method before an exam. The calculator shows the full working, not just the final answer.

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