Integral Calculator

Integral Calculator

Indefinite Integral Calculation

Integrand f(x):

Examples: `3*x^2 - 2*x`, `x^4 + sin(x) + cos(x)`, `exp(x)`

Definite Integral Calculation

Integrand f(x):

Lower Limit a:

Upper Limit b:

Examples: `0`, `pi`, `e`, `1/2`, `sqrt(2)`

Area Enclosed by Two Curves

Please enter two curves $y = f(x)$ and $y = g(x)$, and the integration interval $a \le x \le b$.

Curve f(x):

Curve g(x):

Start Point a:

End Point b:

Examples: `0`, `pi`, `e`, `1/2`

💡 Learning Tips: Integration & Area

Indefinite Integral: The reverse operation of differentiation, which finds the antiderivative $F(x) + C$ of the function ($C$ is the integration constant).
- Power Rule: $\int x^n dx = \frac{1}{n+1} x^{n+1} + C \quad (n \neq -1)$
Definite Integral: Evaluates the difference $F(b) - F(a)$ by substituting the limits into the antiderivative, representing the signed area between the graph and the $x$-axis.
Area Between Two Curves: If $f(x) \ge g(x)$ in the interval, the enclosed area $S$ is represented by: $$S = \int_{a}^{b} (f(x) - g(x)) dx$$ If the curves intersect and change order, find the intersection points to split the interval or integrate the absolute difference.