What is an indefinite integral?

An indefinite integral is the antiderivative of a function — a function F(x) whose derivative is f(x). ∫f(x) dx = F(x) + C The + C (constant of integration) is essential because any constant added to F(x) also has derivative f(x). There are infinitely many antiderivatives, all differing by a constant. Example: ∫2x dx = x² + C Verification: d/dx(x² + C) = 2x ✓ Contrast with definite integral ∫[a to b] f(x) dx, which gives a specific number (the area under the curve from a to b) using the Fundamental Theorem of Calculus: ∫[a,b] f(x) dx = F(b) - F(a).

How do you apply the power rule for integration?

The power rule for integration is the reverse of the derivative power rule: ∫x^n dx = x^(n+1) / (n+1) + C (for n ≠ -1) Steps: add 1 to the exponent, then divide by the new exponent. Examples: - ∫x^3 dx = x^4/4 + C - ∫x^10 dx = x^11/11 + C - ∫x dx = x^2/2 + C (n=1) - ∫1 dx = x + C (n=0, since x^0=1) - ∫x^(-2) dx = x^(-1)/(-1) + C = -1/x + C - ∫√x dx = ∫x^(1/2) dx = x^(3/2)/(3/2) + C = (2/3)x^(3/2) + C For n = -1: ∫x^(-1) dx = ∫1/x dx = ln|x| + C (the exception) For polynomials, apply term by term: ∫(3x^2 + 4x - 5) dx = x^3 + 2x^2 - 5x + C

What is the difference between definite and indefinite integrals?

Indefinite integral: ∫f(x) dx = F(x) + C - Result is a function (family of antiderivatives) - Always includes + C - Represents the general antiderivative - This calculator computes indefinite integrals Definite integral: ∫[a to b] f(x) dx = F(b) - F(a) - Result is a number - No + C (it cancels out) - Represents the net signed area under the curve from a to b - Computed using the Fundamental Theorem of Calculus Connection: to evaluate a definite integral, first find the indefinite integral F(x), then subtract: F(b) - F(a). Example: ∫[0 to 2] x^2 dx - Indefinite: F(x) = x^3/3 + C - Definite: F(2) - F(0) = 8/3 - 0 = 8/3

Why is ∫1/x dx = ln|x| and not ln(x)?

The absolute value is needed because ln is only defined for positive numbers, but 1/x exists for both positive and negative x (just not at x=0). For x > 0: d/dx(ln x) = 1/x ✓ For x 0: ln|x| = ln(x) - x < 0: ln|x| = ln(-x) In either case, d/dx(ln|x|) = 1/x. Practically: in most calculus courses and this calculator, you will see both ln(x) and ln|x|. When working with definite integrals on a positive domain, ln(x) is fine. The absolute value matters when x might be negative.

How do you verify an integral answer?

Always verify by differentiating your answer. The derivative of F(x) should equal the original f(x). F(x) = antiderivative, so d/dx[F(x)] = f(x) Examples: - F(x) = x^4/4 + C → d/dx = x^3 = f(x) ✓ (original was x^3) - F(x) = -cos(x) + C → d/dx = sin(x) = f(x) ✓ - F(x) = e^x + C → d/dx = e^x = f(x) ✓ Use the Derivative Calculator on this site to check automatically: 1. Get the integral F(x) from this calculator 2. Enter F(x) into the Derivative Calculator 3. The result should equal your original f(x) If it does not match, check: correct rule applied, correct simplification, signs correct (especially for trig), exponents correct.

Integral Calculator — Step-by-Step Symbolic Integration

The Integral Calculator computes indefinite integrals symbolically with full step-by-step working. Supports polynomials (power rule), trig integrals (sin, cos, tan, sec, csc, cot), exponentials (eˣ), natural logarithm (ln x), square roots, and linear substitution for functions like sin(2x). Every st...

INTEGRAL CALCULATOR

∫ f(x) dx = ? (indefinite integral)

Use * for multiplication, ^ for powers, exp(x) for eˣ. Press Enter or click ∫ dx.

QUICK EXAMPLES

Enter a function above and press ∫ dx or Enter

INTEGRATION RULES

Constant rule∫c dx = cx + C

Power rule∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1)

Reciprocal∫1/x dx = ln|x| + C

Constant mult.∫c·f dx = c·∫f dx

Sum rule∫(f+g) dx = ∫f dx + ∫g dx

∫sin x dx-cos x + C

∫cos x dxsin x + C

∫tan x dx-ln|cos x| + C

∫sec x dxln|sec x + tan x| + C

∫csc x dx-ln|csc x + cot x| + C

∫cot x dxln|sin x| + C

∫eˣ dxeˣ + C

∫aˣ dxaˣ/ln(a) + C

∫ln x dxx·ln(x) - x + C

∫√x dx(2/3)x^(3/2) + C

∫arctan x dxx·arctan(x) - ln(1+x²)/2 + C

INPUT SYNTAX

x^3x³ (power)

2*x2x (multiplication)

sin(x)sine of x

exp(x)eˣ

ln(x)natural log

sqrt(x)√x

sin(2*x)sin(2x) — linear substitution

Created with❤️byeaglecalculator.com

INTEGRATION FORMULAS

INTEGRAL RULES REFERENCE

RULE	FORMULA	DESCRIPTION
Constant rule	∫c dx = cx + C	Integral of any constant
Power rule	∫x^n dx = x^(n+1)/(n+1) + C	n can be any real number except -1
Reciprocal	∫1/x dx = ln\|x\| + C	Special case of power rule at n = -1
Sum rule	∫(f+g) dx = ∫f dx + ∫g dx	Integrate term by term
Constant mult.	∫c·f dx = c·∫f dx	Pull constants outside the integral
∫sin x dx	-cos x + C
∫cos x dx	sin x + C
∫tan x dx	-ln\|cos x\| + C
∫e^x dx	e^x + C	Unique: integral equals itself
∫a^x dx	a^x / ln(a) + C
∫ln x dx	x·ln(x) - x + C	Integration by parts: u=ln(x), dv=dx
∫√x dx	(2/3)x^(3/2) + C	Power rule with n=1/2
∫arctan x dx	x·arctan(x) - ln(1+x²)/2 + C	Integration by parts

HOW TO USE

1
Type your function using standard notation: x^3 for x³, 2*x for 2x, sin(x), cos(x), exp(x) for eˣ, ln(x) for natural log, sqrt(x) for √x. Press Enter or click ∫ dx.
2
The indefinite integral appears as F(x) + C. The constant of integration C is always included — this is what makes it indefinite. Every valid antiderivative differs only by a constant.
3
The step-by-step panel shows which rule was applied: constant rule, power rule, sum/difference rule, constant multiple, or specific trig and exponential rules.
4
Use the quick example buttons: x³ (power rule), 3x²+4x-5 (polynomial), sin(x) and cos(x) (trig), eˣ (exponential), 1/x (reciprocal → ln), √x (fractional power), ln(x) (by parts), sin(2x) (linear substitution).
5
If the function cannot be integrated symbolically (e.g. products of two variable functions like x*sin(x)), the calculator shows a clear error with guidance. Use the Derivative Calculator to verify your result: the derivative of F(x) should equal f(x).

WORKED EXAMPLE

∫(3x^2 + 4x - 5) dx: sum rule applied term by term. ∫3x^2 dx = x^3 (constant multiple + power rule: 3 * x^3/3). ∫4x dx = 2x^2 (power rule). ∫-5 dx = -5x (constant rule). Result: x^3 + 2x^2 - 5x + C. Verification: d/dx(x^3+2x^2-5x+C) = 3x^2+4x-5 ✓. Also: ∫sin(2x) dx = -cos(2x)/2 + C (linear substitution).

FREQUENTLY ASKED QUESTIONS

RELATED CALCULATORS

Derivative Calculator

Calculate instantly →

MORE CALCULUS CALCULATORS

Derivative Calculator

Calculate instantly →

Was this calculator helpful?

Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.