Integration Substitution Calculator

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Expertly Reviewed by: David Chen, CFA & Mathematics Researcher

Verified for mathematical accuracy and pedagogical clarity on Integration Techniques.

Mastering Calculus becomes significantly easier with our Integration Substitution Calculator. This tool helps you identify the correct u-substitution components, calculates the derivative $du$, and simplifies complex integrals into manageable forms instantly.

Integration Substitution Calculator

Enter the inner function you want to substitute.
Enter the function in terms of u.
Assumed standard differential unit.
Resulting Integral Substitution:

Integration Substitution Calculator Formula

$$\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du$$

Where $u = g(x)$ and $du = g'(x) \, dx$.

Source: Wolfram MathWorld – Substitution Rule

Variables:

  • $u$ (Substitution): The chosen inner function $g(x)$ that simplifies the integral.
  • $f(u)$ (Outer Function): The main function structure expressed using the new variable $u$.
  • $du$ (Differential): The derivative of $u$ with respect to $x$, multiplied by $dx$.
  • $C$ (Constant): The arbitrary constant added to all indefinite integrals.

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What is an Integration Substitution Calculator?

The Integration Substitution Calculator is a specialized mathematical tool designed to automate the “U-Substitution” method in calculus. This method is the reverse of the chain rule in differentiation. It allows mathematicians to transform a difficult integral into a simpler one by changing the variable of integration.

By defining a part of the integrand as $u$ and finding its derivative $du$, you can often cancel out complex terms, making the integration process straightforward and less prone to manual errors.

How to Calculate Integration Substitution (Example)

  1. Identify $u$: Look for a function within the integral whose derivative is also present. Example: $\int 2x \cos(x^2) dx$, let $u = x^2$.
  2. Find $du$: Differentiate $u$ with respect to $x$. Here, $du/dx = 2x$, so $du = 2x dx$.
  3. Substitute: Replace $x^2$ with $u$ and $2x dx$ with $du$. The integral becomes $\int \cos(u) du$.
  4. Integrate: The integral of $\cos(u)$ is $\sin(u) + C$.
  5. Back-Substitute: Replace $u$ with the original $x$ expression: $\sin(x^2) + C$.

Frequently Asked Questions (FAQ)

When should I use the substitution method? Use it when an integrand contains a function and its derivative, or when a change of variable significantly simplifies the expression.

Can $u$ be any function? Yes, but choosing a $u$ whose derivative $du$ helps cancel other parts of the integrand is the key to success.

What if the derivative is off by a constant? You can adjust for constants by dividing or multiplying outside the integral sign.

Is u-substitution used for definite integrals? Absolutely. Just remember to either change the limits of integration to match $u$ or back-substitute before applying the original limits.

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