Calculate definite and indefinite integrals with step-by-step solutions
x^2, 2*x, 1/x, sqrt(x), pisin(x), cos(x), tan(x), asin(x)e^x, exp(x), ln(x), log(x,b)sinh(x), cosh(x), tanh(x)
| ∫xndx | = xn+1/(n+1) + C |
| ∫exdx | = ex + C |
| ∫1/x dx | = ln|x| + C |
| ∫sin(x)dx | = -cos(x) + C |
| ∫cos(x)dx | = sin(x) + C |
An integral is the reverse operation of differentiation (finding antiderivatives). Geometrically, a definite integral represents the area under a curve between two points. In physics, integrals represent accumulation: distance is the integral of velocity, and work is the integral of force.
∫f(x)dx = F(x) + C
Represents a family of functions whose derivative is f(x). The constant C accounts for all possible vertical shifts.
∫ab f(x)dx = F(b) - F(a)
Represents the exact area under the curve f(x) from x=a to x=b. Uses the Fundamental Theorem of Calculus.
Part 1: If F(x) = ∫ax f(t)dt, then F'(x) = f(x)
Part 2: If F'(x) = f(x), then ∫ab f(x)dx = F(b) - F(a)
This theorem connects differentiation and integration as inverse operations, providing a systematic way to evaluate definite integrals.
∫xndx = xn+1/(n+1) + C (n ≠ -1)
Example: ∫x³dx = x⁴/4 + C
Let u = g(x), then du = g'(x)dx
Example: ∫2x·cos(x²)dx, let u = x², du = 2x dx → ∫cos(u)du = sin(u) + C = sin(x²) + C
∫u dv = uv - ∫v du
Example: ∫x·exdx, let u = x, dv = exdx → x·ex - ∫exdx = x·ex - ex + C
Decompose rational functions into simpler fractions
Example: ∫1/(x²-1)dx = ∫[1/2(1/(x-1)) - 1/2(1/(x+1))]dx
Before learning the Fundamental Theorem, integrals were approximated using Riemann sums - dividing the area into rectangles and summing their areas:
Uses left endpoint of each subinterval
Uses right endpoint of each subinterval
Uses middle of each subinterval (more accurate)
Uses trapezoids instead of rectangles
As the number of rectangles approaches infinity (width → 0), the Riemann sum converges to the exact integral value. Try the interactive Riemann sum visualization above to see this in action!
Find areas between curves, areas of irregular shapes, and regions bounded by multiple functions.
Calculate volumes by rotating curves around axes using disk, washer, or shell methods.
Work = ∫F·dx, Center of Mass, Fluid Pressure, Arc Length, and more.
Consumer/Producer Surplus, Total Revenue from Marginal Revenue, Present Value of Income Streams.
Probability density functions, Expected values, Normal distribution calculations.
Total distance from velocity, Total growth from rate of change, Net change problems.
Explore more calculus tools for complete mathematical analysis.
Every coffee helps keep the servers running. Every book sale funds the next tool I'm dreaming up. You're not just supporting a site — you're helping me build what developers actually need.