How to Use the Power Rule of Integration in Calculus

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The power rule of integration gives you the general solution for the integral of any variable raised to any power except -1, which represents a special case. Since integrals are antiderivatives -- in other words, if you integrate the derivative of a function, you end up with the original function -- think of the power rule of integration as doing the opposite of what the power rule for derivatives does.

Things You'll Need

  • Paper
  • Pencil
  • Convert any square roots, roots of other powers and powers in denominators to standard power functions. The square root of x equals x^(1/2), the cube root of x equals x^(1/3) and so on for the other roots. To move a power from the denominator to the numerator, take the inverse of the power: 1/x^2 = x^-2, for example.

  • Add one to the power. For int[(x^3)dx], for example, x^3 becomes x^4.

  • Divide the result by the new power. For example, x^4 becomes (x^4)/4.

  • Add the constant of integration, usually represented by c, to complete your answer. For example, [(x^4)/4] + c.

Tips & Warnings

  • To integrate a constant, think of the constant as being multiplied by x^0. For example, int(2 dx) = int[(2x^0)dx] = (2x^1)/1 + c = 2x + c.
  • If the integral includes addition or subtraction, integrate each part of the function separately; think of int[(x + 2)dx] as int(x dx) + int (2 dx), for example.
  • The integral of 1/x, or x^-1, equals ln|x| + c.
  • When working with negative exponents, remember that adding one will make the absolute value of the exponent smaller; x^-3 becomes x^-2, not x^-4.

References

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