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Write The Integral in Terms of U Calculator

Reviewed by Calculator Editorial Team

This guide explains how to rewrite integrals in terms of u using substitution. The calculator on this page helps you perform u-substitution for both definite and indefinite integrals.

What is u-substitution?

U-substitution is a technique used in calculus to simplify integrals by substituting a part of the integrand with a new variable, typically u. This method is particularly useful when the integrand is a composite function.

The general form of u-substitution is:

Let u = g(x), then du = g'(x)dx

∫f(x)dx = ∫f(g(u)) * g'(u) du

The key steps in u-substitution are:

  1. Identify a suitable substitution u = g(x)
  2. Find the derivative du = g'(x)dx
  3. Rewrite the integral in terms of u
  4. Integrate with respect to u
  5. Substitute back in terms of x if needed

U-substitution is particularly effective when the integrand contains a function and its derivative, such as e^x, sin(x), or x^n.

How to use the calculator

The calculator on this page helps you perform u-substitution for integrals. Here's how to use it:

  1. Enter the integrand in the "Integrand" field
  2. Specify the variable of integration (usually x)
  3. Enter your substitution choice (usually u)
  4. Click "Calculate" to see the rewritten integral

The calculator will show you the rewritten integral in terms of your substitution variable, along with the steps and assumptions used.

Note: The calculator assumes you've already identified a suitable substitution. It doesn't automatically determine the best substitution for you.

Steps for u-substitution

Follow these steps to perform u-substitution:

  1. Identify the substitution

    Choose a substitution u that simplifies the integral. Common choices include:

    • u = x^n (for polynomials)
    • u = e^x (for exponential functions)
    • u = sin(x) or cos(x) (for trigonometric functions)

  2. Find the derivative

    Compute du/dx and express it as du = g'(x)dx

  3. Rewrite the integral

    Replace x with u and dx with du/g'(x)

  4. Integrate

    Integrate the rewritten expression with respect to u

  5. Substitute back

    Convert the result back in terms of the original variable if needed

Examples

Here are some examples of u-substitution:

Example 1: Polynomial integral

Integrand: x^2 * (x^3 + 1)^5

Substitution: u = x^3 + 1

Result: (1/3)(x^3 + 1)^6 + C

Example 2: Trigonometric integral

Integrand: sin(x) * cos(x)^3

Substitution: u = cos(x)

Result: (1/4)cos(x)^4 + C

Example 3: Exponential integral

Integrand: e^x * sin(e^x)

Substitution: u = e^x

Result: (1/2)(1 - cos(e^x)) + C

FAQ

When should I use u-substitution?
Use u-substitution when the integrand contains a composite function that, when substituted, simplifies the integral.
What if I can't find a suitable substitution?
If you can't find a substitution that simplifies the integral, try other techniques like integration by parts or trigonometric identities.
Can I use u-substitution for definite integrals?
Yes, u-substitution works for definite integrals. You'll need to adjust the limits of integration accordingly.
What if the substitution leads to a complex integral?
If the substitution makes the integral more complex, it's likely not the best choice. Try a different substitution.
How do I know if I've done u-substitution correctly?
Check that you've correctly identified u, found du, and rewritten the integral properly. The result should be simpler than the original.