Integral Calculator

Calculate integrals with step-by-step solutions. Supports definite and indefinite integrals, power rule, trigonometric integrals, and more.

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Integration Practice Problems

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Practice with 5 problems to test your understanding.

What is an Integral?

An integral is a fundamental concept in calculus that represents the accumulation of quantities. It can be thought of as the reverse of differentiation (finding the antiderivative) or as calculating the area under a curve.

There are two types of integrals: indefinite integrals (antiderivatives) which give a family of functions, and definite integrals which give a numerical value representing the signed area between a function and the x-axis over an interval.

Key Facts About Integrals

  • •The integral of x^n is x^(n+1)/(n+1) + C (Power Rule)
  • •The integral of 1/x is ln|x| + C
  • •The integral of e^x is e^x + C
  • •The integral of sin(x) is -cos(x) + C
  • •The integral of cos(x) is sin(x) + C
  • •Always add the constant of integration (+C) for indefinite integrals
  • •Definite integrals use the Fundamental Theorem of Calculus: F(b) - F(a)
  • •Integration is the inverse operation of differentiation

Frequently Asked Questions

What is the integral of x^2?
Using the power rule, the integral of x^2 is x^3/3 + C. Add 1 to the exponent (2+1=3) and divide by the new exponent.
What is the difference between definite and indefinite integrals?
An indefinite integral gives you a family of antiderivatives (with +C), while a definite integral gives you a specific numerical value by evaluating the antiderivative at two bounds.
Why do we add +C to indefinite integrals?
The constant of integration C represents all possible vertical shifts of the antiderivative. Since the derivative of any constant is zero, we must account for this unknown constant.
What is u-substitution?
U-substitution is a technique where you substitute u = g(x) to simplify composite functions. It's essentially the chain rule in reverse: if du = g'(x)dx, the integral becomes simpler.

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Formula:
By Joseph OrdunaHow this works

How this works

Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.

Sources

  1. [1]
    NIST Digital Library of Mathematical Functions
    Academicdlmf.nist.govAccessed Apr 21, 2026
Joseph Orduna
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Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).

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