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Math

Integral Calculator

Solve indefinite and definite integrals with symbolic computation. See the result in LaTeX, the decimal value and the area-under-the-curve plot.

Table of integrals

The most common basic integrals, with the constant of integration C. They are the starting point of every solution by substitution or by parts.

IntegralResult
∫k dx\int k\,dxkx+Ckx + C
∫xn dx  (n≠−1)\int x^n\,dx \;(n \neq -1)xn+1n+1+C\frac{x^{n+1}}{n+1} + C
∫1x dx\int \frac{1}{x}\,dxln⁡∣x∣+C\ln|x| + C
∫ex dx\int e^x\,dxex+Ce^x + C
∫ax dx\int a^x\,dxaxln⁡a+C\frac{a^x}{\ln a} + C
∫ln⁡x dx\int \ln x\,dxxln⁡x−x+Cx\ln x - x + C
∫sin⁡x dx\int \sin x\,dx−cos⁡x+C-\cos x + C
∫cos⁡x dx\int \cos x\,dxsin⁡x+C\sin x + C
∫tan⁡x dx\int \tan x\,dx−ln⁡∣cos⁡x∣+C-\ln|\cos x| + C
∫sec⁡2x dx\int \sec^2 x\,dxtan⁡x+C\tan x + C
∫11+x2 dx\int \frac{1}{1+x^2}\,dxarctan⁡x+C\arctan x + C
∫11−x2 dx\int \frac{1}{\sqrt{1-x^2}}\,dxarcsin⁡x+C\arcsin x + C
∫1a2+x2 dx\int \frac{1}{a^2+x^2}\,dx1aarctan⁡xa+C\frac{1}{a}\arctan\frac{x}{a} + C

Properties and methods

  • ∫(f±g) dx=∫f dx±∫g dx\int (f \pm g)\,dx = \int f\,dx \pm \int g\,dx
  • ∫c f dx=c∫f dx\int c\,f\,dx = c\int f\,dx
  • ∫f(g(x)) g′(x) dx=∫f(u) du\int f(g(x))\,g'(x)\,dx = \int f(u)\,du
  • ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
  • ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

Frequently Asked Questions

Clear your doubts and see practical examples

Use standard math syntax. Valid examples: sin(x)*x, 1/(1+x^2), e^x, x^2 + 3*x - 1, sqrt(x), ln(x). Use * for multiplication and ^ for power.

The indefinite integral (antiderivative) returns a family of functions F(x) + C whose derivative is f(x). The definite integral computes the net area under the curve between two limits a and b, returning a number.

Yes. The symbolic engine applies rules like substitution, integration by parts, partial fractions and trigonometric integrals. For cases without an elementary antiderivative, we compute numerically with Simpson's rule using 1000 subintervals.

Every indefinite integral has infinitely many solutions that differ by a constant. The "+ C" represents that arbitrary constant of integration.

Yes. Use pi for π and e for the natural base. Example: integrate cos(x) from 0 to pi/2, or e^x.