Integral Calculator – Solve Integrals Instantly
Calculate definite and indefinite integrals with step-by-step solutions
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Integration Rules
∫cos(x) dx = sin(x) + C
Integration Results
What is Integral Calculus?
Integration is one of the two main operations of calculus, essentially acting as the inverse of differentiation to find totals.
Area Under Curves
The most common visual interpretation of a definite integral is the total area between the function's graph and the x-axis.
Reverse Derivative
Integration is often called "antidifferentiation." If taking a derivative tells you the slope, integrating tells you the original function.
The Notation ∫
The integral symbol (∫) is an elongated "S", representing "Summation", as integration adds up infinitely many infinitesimal slices.
Definite Integrals
A definite integral has start and end points (limits). It calculates a specific numerical value representing accumulated change.
Indefinite Integrals
An indefinite integral represents a family of functions. The result always includes a "+ C" to account for unknown constants.
Fundamental Theorem
This theorem links differentiation and integration, allowing us to compute definite integrals using antiderivatives.
Where is Integration Used?
From calculating the volume of a sphere to determining the work done by a force, integration is vital in science and engineering.
Physics & Motion
Integrating an object's velocity function over time gives you its displacement (position change), and integrating acceleration gives velocity.
Center of Mass
Engineers use integration to find the centroid or center of mass of irregular shapes, crucial for stability in construction.
Economics
Economists integrate "marginal cost" functions to determine the total cost of production or to calculate consumer surplus.
Volume Calculation
Integrals calculate the volume of 3D objects created by rotating a 2D curve around an axis (Solids of Revolution).
Probability
In statistics, the area under a probability density function (calculated via integration) determines the probability of an event.
Electronics
Integration is used to calculate the voltage across a capacitor by integrating the current flow over time.
How to Solve Integrals?
While calculators are great, understanding these key techniques and rules helps you solve complex calculus problems manually.
The Power Rule
The reverse of the derivative power rule. Increase the exponent by 1, then divide by the new exponent: ∫xⁿ dx = xⁿ⁺¹/(n+1).
Don't Forget +C
For indefinite integrals, always add the Constant of Integration (+C). It represents the infinite vertical shifts of the function.
U-Substitution
The "Reverse Chain Rule." Substitute a complex part of the function with 'u' to simplify the integral into a basic form.
Integration by Parts
The "Reverse Product Rule." Used when integrating the product of two functions. Formula: ∫udv = uv - ∫vdu.
Linearity Property
You can split integrals of sums into sums of integrals: ∫(f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx. Constants move outside.
Trig Identities
Memorize your trig integrals! For example, the integral of cos(x) is sin(x), but the integral of sin(x) is -cos(x).
