Calculators / Calculus 2 / Integration by Parts

Integration by Parts Calculator

Published on: May 25, 2025

This Integration by Parts Calculator helps you solve integrals using the integration by parts method and shows each step clearly. It works by choosing suitable parts for u and dv, differentiating and integrating them, and then applying the standard formula correctly. This makes it useful for checking answers, understanding how the method works, and practising calculus step by step.

Step-by-step method

  1. Write the integral.
  2. Choose u and dv (LIATE heuristic).
  3. Compute du and v.
  4. Use the integration-by-parts formula.
  5. Apply the formula.
  6. Solve the remaining integral.
  7. Simplify and add + C.

Formula:

∫ u dv = u·v − ∫ v du

Example 1: x*e^x, x

Step 1 - Write the integral.

In this problem: We will apply integration by parts.

∫xexdx

Step 2 - Pick u and dv.

In this problem: Using LIATE, choose u to differentiate easily.

u = xdv = ex dx

Step 3 - Compute du and v.

In this problem: Differentiate u and integrate dv.

du = 1 dxv = ∫ ex dx = ex

Step 4 - Formula.

In this problem: Integration by parts identity:

∫ u dv = u·v − ∫ v du

Step 5 - Apply the formula.

In this problem: Substitute u, v, du into the identity.

∫xexdx = x·ex − ∫ (ex)(1) dx

Step 6 - Solve the remaining integral.

In this problem: Compute the leftover integral.

∫ (ex)(1) dx = ex

Step 7 - Combine and simplify.

In this problem: Substitute back, simplify, and add the constant.

x·ex − (ex) = (x − 1)ex + C

Final answer: (x - 1)exp(x) + C

Example 2: x*sin(x), x

Step 1 - Write the integral.

In this problem: We will apply integration by parts.

∫xsin(x)dx

Step 2 - Pick u and dv.

In this problem: Using LIATE, choose u to differentiate easily.

u = xdv = sin(x) dx

Step 3 - Compute du and v.

In this problem: Differentiate u and integrate dv.

du = 1 dxv = ∫ sin(x) dx = −cos(x)

Step 4 - Formula.

In this problem: Integration by parts identity:

∫ u dv = u·v − ∫ v du

Step 5 - Apply the formula.

In this problem: Substitute u, v, du into the identity.

∫xsin(x)dx = x·−cos(x) − ∫ (−cos(x))(1) dx

Step 6 - Solve the remaining integral.

In this problem: Compute the leftover integral.

∫ (−cos(x))(1) dx = −sin(x)

Step 7 - Combine and simplify.

In this problem: Substitute back, simplify, and add the constant.

x·−cos(x) − (−sin(x)) = −xcos(x) + sin(x) + C

Final answer: -xcos(x) + sin(x) + C