Antiderivatives and Basic Integrals

Reverse the power rule of differentiation: raise the exponent by 1, then divide by the new exponent. xⁿ becomes x^(n+1)/(n+1) + C, valid for all n ≠ −1. Constant factors stay, sums are handled term by term. Example: f(x) = 4x³ − 2x + 5 has antiderivatives F(x) = x⁴ − x² + 5x + C. The rule also works for negative and fractional exponents: ∫x⁻² dx = −x⁻¹ + C and ∫√x dx = ∫x^(1/2) dx = (2/3)x^(3/2) + C. Safest check: differentiate the result, it must give exactly the integrand. Only the case n = −1 drops out; there the antiderivative is ln|x| + C.

Because the derivative of every constant is zero. If F is an antiderivative of f, then so are F + 7 or F − 123, since the constant vanishes without trace when differentiating. The set of all antiderivatives is therefore a whole family of vertically shifted graphs, and the +C stands for these infinitely many possibilities. C is pinned down only by an extra condition, such as "the graph passes through (0|1)": for f(x) = e^(2x), F(x) = ½e^(2x) + C, and F(0) = 1 forces C = ½. In a definite integral, however, C cancels in F(b) − F(a), so there you may drop it.

The core set is short: the power rule ∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1, the special case ∫1/x dx = ln|x| + C, the exponential ∫eˣ dx = eˣ + C with the variant ∫e^(kx) dx = (1/k)e^(kx) + C, and the trigonometric pair ∫sin x dx = −cos x + C and ∫cos x dx = sin x + C. Add the structural rules: integrate term by term, pull constant factors out front, and for an inner linear function f(ax + b) divide by a. With this toolbox you cover almost all mandatory-part integrals; more complicated products are handled by integration by parts. Regular self-check: differentiate backwards.

In German classrooms "aufleiten" is student slang for finding an antiderivative, i.e. reversing differentiation; the proper term is indefinite integration. Mathematics distinguishes two things precisely: the indefinite integral ∫f(x) dx denotes the set of all antiderivatives F + C. The definite integral ∫ₐᵇ f(x) dx, by contrast, is a number, defined as a limit of rectangle sums, measuring signed areas. The fundamental theorem connects both worlds: you obtain the number by evaluating any antiderivative at the limits, F(b) − F(a). In exams use the term antiderivative; "aufleiten" is understood by everyone but counts as colloquial.

Use the linear substitution rule: if F is an antiderivative of f, then ∫f(ax + b) dx = (1/a)·F(ax + b) + C. The inner linear function stays, and you divide by its slope a instead of multiplying by it as in differentiation. Examples: ∫e^(3x) dx = (1/3)e^(3x) + C, ∫sin(2x) dx = −(1/2)cos(2x) + C, ∫(4x + 1)⁵ dx = (1/4)·(4x + 1)⁶/6 + C. Checking by differentiating shows why: the chain rule produces the factor a, which the 1/a cancels again. Caution: this convenient rule holds ONLY for linear inner functions; for ∫e^(x²) dx it fails, such an integral has no elementary antiderivative.