U Du Integral Calculator

This U du Integral Calculator helps you compute integrals of the form ∫u du. Learn how to calculate u du integrals, understand the formula, and see practical examples.

What is a u du integral?

A u du integral is a basic type of indefinite integral where the integrand is a function u(x) multiplied by its derivative du. This type of integral is fundamental in calculus and appears in many applications, including physics, engineering, and economics.

The integral ∫u du is a special case of integration by substitution, where the substitution is u itself. This makes the integral straightforward to solve once you understand the relationship between u and du.

How to calculate u du integral

Calculating a u du integral involves recognizing that the integral of u with respect to x is equal to the antiderivative of u(x). The key step is to express the integral in terms of u rather than x.

Steps to solve ∫u du

Identify the function u(x) and its derivative du/dx.
Express the integral in terms of u: ∫u du.
Integrate u with respect to u: (u²/2) + C, where C is the constant of integration.

This method works because the derivative of u with respect to x is du/dx, and the integral of u with respect to x is the antiderivative of u(x).

Formula

The formula for the u du integral is:

∫u du = (u²/2) + C

Where:

u is a function of x
du is the differential of u with respect to x
C is the constant of integration

This formula is derived from the fundamental theorem of calculus, which states that the integral of a function is its antiderivative.

Example

Let's solve the integral ∫(2x + 3) dx using the u du method.

Step 1: Let u = 2x + 3

We set u equal to the integrand: u = 2x + 3.

Step 2: Find du

Differentiate u with respect to x to find du: du = 2 dx.

Step 3: Rewrite the integral

Express the integral in terms of u: ∫(2x + 3) dx = ∫u (du/2).

Step 4: Integrate

Now integrate u with respect to u: (1/2)∫u du = (1/2)(u²/2) + C = u²/4 + C.

Step 5: Substitute back

Replace u with the original expression: u²/4 + C = (2x + 3)²/4 + C.

The final result is (2x + 3)²/4 + C.

FAQ

What is the difference between ∫u du and ∫u dx?

∫u du is an integral where the integrand is u and the differential is du, which is the derivative of u with respect to x. ∫u dx is an integral where the integrand is u and the differential is dx. The key difference is that ∫u du is expressed in terms of u, while ∫u dx is expressed in terms of x.

When should I use the u du integral method?

You should use the u du integral method when the integrand is a function u(x) and its derivative du/dx is known. This method simplifies the integral by expressing it in terms of u rather than x.

What is the constant of integration C?

The constant of integration C represents the family of curves that have the same derivative. It is necessary because indefinite integrals have infinitely many solutions, differing only by a constant.