Indefinite Integral Calculator

Math Calculus • Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

1. Indefinite Integral Calculator

Computes an antiderivative \(F(x)\) for \(\int f(x)\,dx\) and plots \(f(x)\), \(F(x)+C\), and \(F'(x)\). Includes a numeric “diff-back” verification on the current window (optional).

Inputs

Integrand \(f(x)\)

Supported: + − * / ^, parentheses, variable x, constants pi, e, sin cos tan, ln log(base 10), sqrt abs exp. Implicit multiplication: 2x, (x+1)(x-1), xsin(x).

Output format

LaTeX is best for readability.

Constant \(C\) (for plotting \(F(x)+C\))

Does not change \(F'(x)\).

Graph center \(c\)

Plots \(x\in[c-w,c+w]\).

Graph window half-width \(w\)

Use Auto fit if scales explode.

Options Show grid Show \(f(x)\) Show \(F(x)+C\) Show \(F'(x)\) Verify by differentiating (numeric check) Show steps / technique

Ready

Graph

Drag to pan • wheel/pinch to zoom • curves: \(f(x)\), \(F(x)+C\), \(F'(x)\)

\(f(x)\) \(F(x)+C\) \(F'(x)\)

x: 0, y: 0, zoom(px/unit): 60

Tip: If the plot scales explode, zoom out or use Auto fit.

Result

Enter \(f(x)\) and click Integrate.

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1. Indefinite Integral Calculator — Theory

An indefinite integral (antiderivative) of a function \(f(x)\) is a function \(F(x)\) such that \(F'(x)=f(x)\). We write:

\[ \int f(x)\,dx = F(x) + C, \]

where \(C\) is an arbitrary constant (all antiderivatives differ by a constant).

Core rules (most common patterns)

\[ \int \big(f(x)\pm g(x)\big)\,dx = \int f(x)\,dx \pm \int g(x)\,dx \] \[ \int k f(x)\,dx = k\int f(x)\,dx \] \[ \int x^n\,dx = \frac{x^{n+1}}{n+1}+C \quad (n\neq -1) \] \[ \int \frac{1}{x}\,dx = \ln|x|+C \] \[ \int \sin(x)\,dx = -\cos(x)+C,\qquad \int \cos(x)\,dx = \sin(x)+C \] \[ \int e^x\,dx = e^x + C \]

Linear inner functions (a basic substitution idea)

If the inside is linear, \(u=ax+b\), then \(du=a\,dx\), so many integrals become “divide by \(a\)”:

\[ \int (ax+b)^n\,dx = \frac{(ax+b)^{n+1}}{a(n+1)}+C \quad (n\neq -1) \] \[ \int \frac{1}{ax+b}\,dx = \frac{1}{a}\ln|ax+b|+C \] \[ \int \sin(ax+b)\,dx = -\frac{1}{a}\cos(ax+b)+C,\qquad \int \cos(ax+b)\,dx = \frac{1}{a}\sin(ax+b)+C \] \[ \int e^{ax+b}\,dx = \frac{1}{a}e^{ax+b}+C \]

Integration by parts

When you see a product (especially “polynomial \(\times\)” trig/exp), use integration by parts:

\[ \int u\,dv = uv - \int v\,du. \]

A standard choice is to let \(u\) be the part that becomes simpler when differentiated (often a polynomial), and \(dv\) be the part that is easy to integrate (like \(\sin\), \(\cos\), \(e^x\)).

Worked example

Compute \(\int x\sin(x)\,dx\).

\[ u=x \Rightarrow du=dx,\qquad dv=\sin(x)\,dx \Rightarrow v=-\cos(x). \] \[ \int x\sin(x)\,dx = uv - \int v\,du = x(-\cos x) - \int (-\cos x)\,dx = -x\cos x + \int \cos x\,dx = -x\cos x + \sin x + C. \]

How to check your answer

The safest habit is to differentiate the result. If \(F'(x)=f(x)\), then \(\int f(x)\,dx = F(x)+C\) is correct (on the domain where the expressions are defined).

Frequently Asked Questions

What does an indefinite integral calculator return?

It returns an antiderivative F(x) such that F'(x) = f(x). The output represents a family of solutions and typically includes a constant of integration C.

Why is there a +C in the indefinite integral result?

Any two antiderivatives of the same function differ by a constant. Adding C accounts for that missing constant because the derivative of a constant is 0.

How does the differentiate verification help?

The calculator can numerically differentiate the computed F(x) over the current graph window and compare it to the original f(x). A close match supports that the antiderivative is consistent with the integrand.

Does changing the constant C affect the derivative curve?

No. Changing C only shifts the plotted F(x)+C up or down, but it does not change F'(x), so the derivative curve remains the same.

What function syntax is supported for the integrand f(x)?

You can use arithmetic operators, powers, parentheses, the variable x, constants like pi and e, and common functions such as sin, cos, tan, ln, log (base 10), sqrt, abs, and exp. Implicit multiplication like 2x or (x+1)(x-1) is also supported.

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