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Types of integrals and how to choose a method

Before choosing a formula, identify what you are calculating. The integral type describes the problem; the method describes how to solve it. A definite integral, for example, may be solved by substitution, by parts or by another technique.

Practice integralsExplore all guides

Classification of integral types

Indefinite integrals

These seek a family of antiderivatives F(x) + C. The result is a function, checked by differentiating on the relevant domain. There are no integration bounds.

∫2x dx=x2+C\int 2x\,dx=x^2+CRead the guide and examples →

Definite integrals

These accumulate a quantity between two bounds. Under the fundamental theorem’s hypotheses, evaluate F(b) − F(a). The value may be negative and is not always a geometric area.

∫02x2 dx=83\int_0^2x^2\,dx=\frac83Read the guide and examples →

Improper integrals

These have infinite bounds or singularities on the interval. They are defined through limits and require convergence checks. An interior singularity requires studying both sides separately.

∫0∞e−x dx=1\int_0^\infty e^{-x}\,dx=1Read the guide and examples →

Double integrals

These integrate over a plane region. Depending on the integrand, they can represent area, the mass of a sheet or volume under a surface. Bounds must describe the full region without counting it twice.

∫01∫01−y1 dx dy=12\int_0^1\int_0^{1-y}1\,dx\,dy=\frac12Read the guide and examples →

Triple integrals

These integrate over a spatial region. An integrand of 1 gives volume; a density gives mass. Each inner bound may depend on variables that have not yet been integrated.

∫01∫01∫011 dz dy dx=1\int_0^1\int_0^1\int_0^1 1\,dz\,dy\,dx=1Read the guide and examples →

Line and surface integrals

These accumulate quantities over curves or surfaces. They need parametrizations, length or area elements and, for circulation or flux, an orientation. Advanced guides explain them; they are not a dedicated calculator mode.

∮∂DP dx+Q dy=∬D(Qx−Py) dA\oint_{\partial D}P\,dx+Q\,dy=\iint_D(Q_x-P_y)\,dARead the guide and examples →

Choosing an integration method

There is no recipe that solves every integral. Start by simplifying the expression and looking for these clues, in this suggested order:

  1. Sums and powers. Apply linearity and immediate antiderivatives. Remember the 1/x exception and its logarithm. Power rule.
  2. A composite function beside its derivative. Try u = g(x) and transform the entire expression. Integration by substitution.
  3. A product where differentiating one factor simplifies it. Choose u and dv, checking that dv can be integrated. Integration by parts.
  4. A quotient of polynomials. Compare degrees. If the numerator degree is at least the denominator degree, divide first. Then factor the denominator. Partial fractions.
  5. Trigonometric powers or radicals. Look for identities or a substitution that simplifies the radical while respecting signs and domains. Trigonometric identities · Trigonometric substitution.

If elementary techniques do not work, special functions or numerical approximation may be needed. The absence of an elementary antiderivative does not mean a definite integral does not exist. Nonelementary antiderivatives.

Polar, cylindrical or spherical: how to decide

A coordinate change should simplify the region or integrand. Rewrite both and use the absolute value of the Jacobian determinant.

  • Polar coordinates: disks, sectors and annuli in the plane. The area element is r dr dθ.
  • Cylindrical coordinates: symmetry around an axis, cylinders, cones and paraboloids. Keep z and use the volume factor r.
  • Spherical coordinates: spheres and radial regions. Here φ is measured from the z axis and the volume factor is r² sin φ.

Always check that the intervals cover the region once. If you change integration order, describe the region again; swapping the differentials is not enough. Changing order and Fubini.

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