Integration by Parts Notes

Published on: September 29, 2026

This Integration by Parts Calculator integrates a product of two functions, and shows the working line by line. One factor is chosen to be differentiated and the other to be integrated, and the formula trades the original integral for an easier one. Choosing the wrong factor makes the new integral harder rather than simpler. Following the same pattern every time keeps that choice deliberate instead of accidental.

Step-by-step method

  1. Set up the problem.
  2. Write the integration by parts formula.
  3. Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).
  4. Differentiate u and integrate dv.
  5. Substitute into the formula.
  6. Evaluate the remaining integral and simplify, adding C.

Formula:

\(\int u \, dv = uv - \int v \, du\)
See Example 1 Hide Example 1

Problem

\(\int x e^{x}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration by parts formula.
  3. Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).
  4. Differentiate u and integrate dv.
  5. Substitute into the formula.
  6. Evaluate the remaining integral and simplify, adding C.

Step 1

Step 1 - Set up the problem.

In this problem: We are integrating a product of two different kinds of functions.

\(\int x e^{x}\, dx\)

Step 2

Step 2 - Write the integration by parts formula.

In this problem: Integration by parts trades one integral for a hopefully easier one.

\(\int u \, dv = uv - \int v \, du\)

Step 3

Step 3 - Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).

In this problem: By LIATE, choose \(u = x\) and \(dv = e^{x} \, dx\).

\(\begin{gathered} u = x \\ dv = e^{x}\, dx \end{gathered}\)

Step 4

Step 4 - Differentiate u and integrate dv.

In this problem: Differentiating u gives \(du = 1 \, dx\); integrating dv gives \(v = e^{x}\).

\(\begin{gathered} du = 1\, dx \\ v = e^{x} \end{gathered}\)

Step 5

Step 5 - Substitute into the formula.

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

\(\int x e^{x}\, dx = \left(x\right)\left(e^{x}\right) - \int e^{x}\, dx\)

Step 6

Step 6 - Evaluate the remaining integral and simplify, adding C.

In this problem: The remaining integral evaluates to \(e^{x}\). Simplifying gives the answer.

\(\int x e^{x}\, dx = x e^{x} - e^{x} + C\)

Final Answer

\(\int x e^{x}\, dx = x e^{x} - e^{x} + C\)
See Example 2 Hide Example 2

Problem

\(\int x \cos{\left(x \right)}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration by parts formula.
  3. Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).
  4. Differentiate u and integrate dv.
  5. Substitute into the formula.
  6. Evaluate the remaining integral and simplify, adding C.

Step 1

Step 1 - Set up the problem.

In this problem: We are integrating a product of two different kinds of functions.

\(\int x \cos{\left(x \right)}\, dx\)

Step 2

Step 2 - Write the integration by parts formula.

In this problem: Integration by parts trades one integral for a hopefully easier one.

\(\int u \, dv = uv - \int v \, du\)

Step 3

Step 3 - Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).

In this problem: By LIATE, choose \(u = x\) and \(dv = \cos{\left(x \right)} \, dx\).

\(\begin{gathered} u = x \\ dv = \cos{\left(x \right)}\, dx \end{gathered}\)

Step 4

Step 4 - Differentiate u and integrate dv.

In this problem: Differentiating u gives \(du = 1 \, dx\); integrating dv gives \(v = \sin{\left(x \right)}\).

\(\begin{gathered} du = 1\, dx \\ v = \sin{\left(x \right)} \end{gathered}\)

Step 5

Step 5 - Substitute into the formula.

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

\(\int x \cos{\left(x \right)}\, dx = \left(x\right)\left(\sin{\left(x \right)}\right) - \int \sin{\left(x \right)}\, dx\)

Step 6

Step 6 - Evaluate the remaining integral and simplify, adding C.

In this problem: The remaining integral evaluates to \(- \cos{\left(x \right)}\). Simplifying gives the answer.

\(\int x \cos{\left(x \right)}\, dx = x \sin{\left(x \right)} + \cos{\left(x \right)} + C\)

Final Answer

\(\int x \cos{\left(x \right)}\, dx = x \sin{\left(x \right)} + \cos{\left(x \right)} + C\)
See Example 3 Hide Example 3

Problem

\(\int x \sin{\left(x \right)}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration by parts formula.
  3. Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).
  4. Differentiate u and integrate dv.
  5. Substitute into the formula.
  6. Evaluate the remaining integral and simplify, adding C.

Step 1

Step 1 - Set up the problem.

In this problem: We are integrating a product of two different kinds of functions.

\(\int x \sin{\left(x \right)}\, dx\)

Step 2

Step 2 - Write the integration by parts formula.

In this problem: Integration by parts trades one integral for a hopefully easier one.

\(\int u \, dv = uv - \int v \, du\)

Step 3

Step 3 - Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).

In this problem: By LIATE, choose \(u = x\) and \(dv = \sin{\left(x \right)} \, dx\).

\(\begin{gathered} u = x \\ dv = \sin{\left(x \right)}\, dx \end{gathered}\)

Step 4

Step 4 - Differentiate u and integrate dv.

In this problem: Differentiating u gives \(du = 1 \, dx\); integrating dv gives \(v = - \cos{\left(x \right)}\).

\(\begin{gathered} du = 1\, dx \\ v = - \cos{\left(x \right)} \end{gathered}\)

Step 5

Step 5 - Substitute into the formula.

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

\(\int x \sin{\left(x \right)}\, dx = \left(x\right)\left(- \cos{\left(x \right)}\right) - \int - \cos{\left(x \right)}\, dx\)

Step 6

Step 6 - Evaluate the remaining integral and simplify, adding C.

In this problem: The remaining integral evaluates to \(- \sin{\left(x \right)}\). Simplifying gives the answer.

\(\int x \sin{\left(x \right)}\, dx = - x \cos{\left(x \right)} + \sin{\left(x \right)} + C\)

Final Answer

\(\int x \sin{\left(x \right)}\, dx = - x \cos{\left(x \right)} + \sin{\left(x \right)} + C\)
See Example 4 Hide Example 4

Problem

\(\int x e^{2 x}\, dx\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration by parts formula.
  3. Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).
  4. Differentiate u and integrate dv.
  5. Substitute into the formula.
  6. Evaluate the remaining integral and simplify, adding C.

Step 1

Step 1 - Set up the problem.

In this problem: We are integrating a product of two different kinds of functions.

\(\int x e^{2 x}\, dx\)

Step 2

Step 2 - Write the integration by parts formula.

In this problem: Integration by parts trades one integral for a hopefully easier one.

\(\int u \, dv = uv - \int v \, du\)

Step 3

Step 3 - Choose u and dv (LIATE: Log, Inverse trig, Algebraic, Trig, Exponential).

In this problem: By LIATE, choose \(u = x\) and \(dv = e^{2 x} \, dx\).

\(\begin{gathered} u = x \\ dv = e^{2 x}\, dx \end{gathered}\)

Step 4

Step 4 - Differentiate u and integrate dv.

In this problem: Differentiating u gives \(du = 1 \, dx\); integrating dv gives \(v = \frac{e^{2 x}}{2}\).

\(\begin{gathered} du = 1\, dx \\ v = \frac{e^{2 x}}{2} \end{gathered}\)

Step 5

Step 5 - Substitute into the formula.

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

\(\int x e^{2 x}\, dx = \left(x\right)\left(\frac{e^{2 x}}{2}\right) - \int \frac{e^{2 x}}{2}\, dx\)

Step 6

Step 6 - Evaluate the remaining integral and simplify, adding C.

In this problem: The remaining integral evaluates to \(\frac{e^{2 x}}{4}\). Simplifying gives the answer.

\(\int x e^{2 x}\, dx = \frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + C\)

Final Answer

\(\int x e^{2 x}\, dx = \frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + C\)