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
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.
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