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What is an integral?

An integral generalizes the concept of summing infinitesimal quantities. It comes in many flavors—∫f(x) dx\int f(x)\,dx (indefinite),∫abf(x) dx\int_{a}^{b} f(x)\,dx (definite),∬Rf(x,y) dA\iint_{R} f(x,y)\,dA (double),∭Vf(x,y,z) dV\iiint_{V} f(x,y,z)\,dV (triple), improper integrals (limits to ∞ or singularities), and line integrals (∮CF⋅dr\oint_{C} \mathbf{F}\cdot d\mathbf{r}).

Integral formula

Indefinite integral (antiderivative):

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

Definite integral:

∫abf(x) dx=F(b)−F(a)\int_{a}^{b} f(x)\,dx = F(b) - F(a)

Double integral:

∬Rf(x,y) dA\iint_{R} f(x,y)\,dA

Triple integral:

∭Vf(x,y,z) dV\iiint_{V} f(x,y,z)\,dV

Improper integral (example):

∫1∞1x2 dx\int_{1}^{\infty} \frac{1}{x^2}\,dx

Line integral:

∮CF(x,y)⋅dr\oint_{C} \mathbf{F}(x,y)\cdot d\mathbf{r}

How to solve an integral step by step

Step 1: Identify f and the integral type—set limits or constant of integration.

Step 2: Choose a method: basic antiderivative rules, substitution, integration by parts, or change to polar/cylindrical/spherical coords for double/triple integrals.

Step 3: Rewrite dVdV or dAdA with the correct Jacobian (rr for polar/cylindrical, ρ2sin⁡ϕ\rho^2\sin\phi for spherical).

Step 4: Integrate the innermost variable first, then proceed outward.

Step 5: For improper integrals, take the limit; for definite, apply the FTC; for line integrals, parametrize the path.

Example: ∫0πsin⁡(x) dx\int_{0}^{\pi} \sin(x)\,dx

∫0πsin⁡(x) dx=[−cos⁡(x)]0π=−cos⁡(π)+cos⁡(0)=−(−1)+1=2 \int_{0}^{\pi} \sin(x)\,dx = [-\cos(x)]_{0}^{\pi} = -\cos(\pi) + \cos(0) = -(-1) + 1 = 2

Tips & Tricks

  • For double/triple integrals, use polar\text{polar}, cylindrical\text{cylindrical}, or spherical\text{spherical} coords when symmetry applies.
  • Use substitution (u ⁣− ⁣subu\!-\!sub) or integration by parts for tricky integrands.
  • Always check convergence in improper integrals by taking limits.
  • When doing line integrals, carefully parametrize CC and compute F⋅dr\mathbf{F}\cdot d\mathbf{r}.

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