Function f(x) =

Use ^ for exponents, * for multiplication, log() = natural log, exp() = e^ , sqrt(). Standard trig: sin, cos, tan, sec, csc, cot.

Evaluate F(c) at x = c (antiderivative with current C)

Constant of integration C (for graph & evaluation)

C = 0.00 drag to shift antiderivative

Compute ∫ & Plot

What is an Indefinite Integral?

In calculus, the indefinite integral (or antiderivative) of a function $f(x)$ is a differentiable function $F(x)$ such that $F^{\prime }(x)=f(x)$. It is written as:

where $C$ represents the constant of integration — any constant whose derivative is zero. Indefinite integrals are fundamental to solving differential equations, computing areas under curves (definite integrals), and modeling physical systems.

Symbolic Integration Engine

Our calculator uses a computer algebra system (nerdamer) to perform exact symbolic integration. It applies power rule, substitution, integration by parts, trigonometric identities, and standard integral tables. Unlike numerical methods, symbolic integration returns closed-form expressions suitable for further analysis. The tool automatically preprocesses ln(x) → log(x) and adds implicit multiplication (2x → 2*x) to reduce syntax errors.

Function $f(x)$	Antiderivative $\int f(x)dx$	Integration Rule
$x^n(n\ne -1)$	$\frac{x^{n+1}}{n+1}+C$	Power Rule
$\frac{1}{x}$	$\ln |x|+C$ (tool returns $\ln (x)+C$ for x>0, mathematically correct for positive domain)	Logarithmic Integral
$e^x$	$e^x+C$	Exponential Rule
$\sin (x)$	$-\cos (x)+C$	Trigonometric
$\cos (x)$	$\sin (x)+C$	Trigonometric
${\sec }^2(x)$	$\tan (x)+C$	Derivative of tan
$\ln (x)$	$x\ln (x)-x+C$	Integration by parts

Real‑world Applications of Indefinite Integrals

• Physics: Determine velocity from acceleration $v(t)=\int a(t)dt$, or position from velocity.
• Economics: Derive total cost from marginal cost functions.
• Engineering: Find deflection curves of beams from load distribution.
• Biology: Model population growth using antiderivatives of growth rates.

How to Use This Calculator: Step-by-Step

1. Enter a mathematical expression in terms of $x$ (e.g., 3*x^2 + sin(x) or 2x + ln(x)).
2. Click Compute Integral — the tool will symbolically integrate the function.
3. The result is displayed in beautiful LaTeX format, including the constant of integration $+C$.
4. A brief description of the computation method is shown (no artificial step breakdown).
5. The graph of the antiderivative (with $C=0$) is plotted over a computed interval.
6. Use example buttons to test classic integrals instantly.

Frequently Asked Questions