Antiderivatives and Basic Integrals

Reverse the power rule of differentiation: raise the exponent by 1, then divide by the new exponent. $x^{n}$ becomes $\frac{x^{(n+1)}}{(n+1)} + C$, valid for all $n \ne{} -1$. Constant factors stay, sums are handled term by term. Example: $f(x) = 4x^{3} - 2x + 5$ has antiderivatives $F(x) = x^{4} - x^{2} + 5x + C$. The rule also works for negative and fractional exponents: $\int{}x^{-2} dx = -x^{-1} + C$ and $\int{}\sqrt{x} dx = \int{}x^{(\frac{1}{2})} dx = (\frac{2}{3})x^{(\frac{3}{2})} + C$. 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 $f(x) = e^{(2x)}$, $F(x) = \frac{1}{2}e^{(2x)} + C$, and $F(0) = 1$ forces $C = \frac{1}{2}$. 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 $\int{}x^{n} dx =\frac{ x^{(n+1)}}{(n+1)} + C$ for $n \ne{} -1$, the special case $\frac{\int{}1}{x} dx = \ln{}|x| + C$, the exponential ∫eˣ dx = eˣ + C with the variant $\int{}e^{(kx)} dx = (\frac{1}{k})e^{(kx)} + C$, and the trigonometric pair $\int{}\sin{} x dx = -\cos{} x + C$ and $\int{}\cos{} x dx = \sin{} x + C$. 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 $\int{}f(ax + b) dx = (\frac{1}{a})\cdot{}F(ax + b) + C$. The inner linear function stays, and you divide by its slope a instead of multiplying by it as in differentiation. Examples: $\int{}e^{(3x)} dx = (\frac{1}{3})e^{(3x)} + C$, $\int{}\sin{}(2x) dx = -(\frac{1}{2})\cos{}(2x) + C$, $\int{}(4x + 1)^{5} dx =\frac{ (\frac{1}{4})\cdot{}(4x + 1)^{6}}{6} + C$. Checking by differentiating shows why: the chain rule produces the factor a, which the $\frac{1}{a}$ cancels again. Caution: this convenient rule holds ONLY for linear inner functions; for $\int{}e^{(x^{2})} dx$ it fails, such an integral has no elementary antiderivative.