Online Integral Solver Calculator

Calculator form

Use the responsive grid below. It shows three columns on large screens, two on smaller screens, and one on mobile.

Mode

Integral type

Variable label

Power coefficient a

Power exponent n

Polynomial coefficients

Enter highest degree first. Example: 2,-3,0,4

Exponential coefficient a

Inner rate b

Trig coefficient a

Frequency b

Phase shift c

Reciprocal numerator a

Denominator slope b

Denominator constant c

Custom expression

Supported functions: sin, cos, tan, exp, log, sqrt, abs, asin, acos, atan, sinh, cosh, tanh.

Lower bound

Upper bound

Decimal precision

Numerical intervals

Graph minimum x

Graph maximum x

Example data table

Mode	Input	Bounds	Expected output
Power	a = 4, n = 3	0 to 2	Antiderivative: x⁴ + C, definite value: 16
Polynomial	2,-3,0,4	-1 to 2	Integrates each term and evaluates F(2) - F(-1)
Exponential	a = 5, b = 2	0 to 1	Antiderivative: 2.5e^2x + C
Custom numeric	x^2 + sin(x)	0 to 3.14159	Composite Simpson estimate with a graph

Formula used

∫ a·xⁿ dx = a·xⁿ⁺¹/(n+1) + C, for n ≠ -1

∫ a·x^-1 dx = a ln|x| + C

∫ a·e^bx dx = (a/b)e^bx + C

∫ a·sin(bx+c) dx = -(a/b)cos(bx+c) + C

∫ a·cos(bx+c) dx = (a/b)sin(bx+c) + C

∫ a/(bx+c) dx = (a/b)ln|bx+c| + C

For custom expressions, the definite result uses composite Simpson’s rule to estimate the area numerically across many even subintervals.

How to use this calculator

Choose the mode that matches your integrand family.
Enter coefficients, exponents, or a custom expression.
Pick indefinite, definite, or both.
Add bounds when you want a definite value.
Set decimal precision, graph range, and numerical intervals.
Press Submit to display the answer above the form.
Review the steps, result table, and graph.
Use the CSV and PDF buttons to export the result.

FAQs

1. Can this solve every integral symbolically?

No. It symbolically solves several common families and numerically estimates custom expressions. That balance keeps the file lightweight, fast, and practical for many classroom and website uses.

2. What custom functions can I type?

You can use sin, cos, tan, exp, log, sqrt, abs, asin, acos, atan, sinh, cosh, and tanh with x, numbers, parentheses, and standard operators.

3. Why does custom mode not show a closed-form antiderivative?

Closed-form symbolic integration for arbitrary expressions requires a much larger algebra system. This version focuses on reliable numeric estimation for custom input and symbolic answers for supported families.

4. What happens when lower and upper bounds are equal?

The definite integral becomes zero because no interval width remains. The calculator also shows a warning so the result is easy to interpret.

5. Why should I increase numerical intervals?

Higher even interval counts usually improve Simpson estimates for smooth functions. They are especially useful when the curve bends often or mixes several changing terms.

6. Can I graph only part of the function?

Yes. Set graph minimum and graph maximum to any range you want. This helps you zoom near turning points, asymptotes, or the definite integration interval.

7. Does the graph shade the definite integral region?

Yes. When definite bounds are present, the plotted region between those bounds is shaded so the selected interval stands out clearly on the graph.

8. What export data is included?

The export captures the mode, integral type, integrand, antiderivative, bounds, definite value, precision, and numerical interval settings in a clean summary table.