Calculator Integrals

Integrals are fundamental in calculus for finding areas under curves, volumes, and solving differential equations. This calculator computes both definite and indefinite integrals with step-by-step solutions and graph visualization.

What is an Integral?

An integral represents the area under a curve between two points. It can be computed as the limit of a Riemann sum. There are two main types:

Definite Integral: Computes the exact area under a curve between two points (a and b).
Indefinite Integral: Finds the antiderivative of a function, representing a family of curves.

Definite Integral Formula:

∫[a to b] f(x) dx = lim(n→∞) Σ[f(xi)Δx], where Δx = (b-a)/n

Integrals have applications in physics, engineering, economics, and probability.

Types of Integrals

Definite Integral

Used to calculate exact areas under curves. For example, the area under y = x² from 0 to 1 is 1/3.

Indefinite Integral

Finds the antiderivative of a function. For example, ∫x² dx = (1/3)x³ + C, where C is the constant of integration.

Improper Integral

Handles infinite limits or discontinuities. For example, ∫[1 to ∞] 1/x² dx converges to 1.

How to Use This Calculator

Select the integral type (definite or indefinite).
Enter the function (e.g., x², sin(x), e^x).
For definite integrals, specify the lower and upper limits.
Click "Calculate" to see the result and graph.

Note: This calculator supports basic functions. Complex integrals may require advanced techniques.

Common Integral Examples

Function	Integral	Result
x²	∫x² dx	(1/3)x³ + C
sin(x)	∫sin(x) dx	-cos(x) + C
e^x	∫e^x dx	e^x + C
1/x	∫1/x dx	ln\|x\| + C

FAQ

What is the difference between definite and indefinite integrals?

Definite integrals compute exact areas between limits, while indefinite integrals find antiderivatives with an arbitrary constant.

Can this calculator solve integrals with trigonometric functions?

Yes, it supports basic trigonometric functions like sin(x) and cos(x).

What if the integral doesn't converge?

The calculator will indicate if the integral diverges (e.g., for 1/x from 0 to 1).