Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation: find indefinite integrals of power, trigonometric, exponential and reciprocal functions.

What this topic is asking

The College Board (Topic 6.8) introduces antiderivatives and indefinite integral notation. An antiderivative reverses differentiation, so the rules are the differentiation rules run backward, with a constant of integration $+C$. You must integrate power, exponential, reciprocal, and basic trigonometric functions.

The basic antiderivative rules

Each rule is checked by differentiating the answer and recovering the integrand. The power rule raises the exponent and divides; the $n = -1$ case is special because dividing by $n + 1 = 0$ fails, and the reciprocal integrates to $\ln|x|$ instead.

A worked indefinite integral

Why the constant of integration is necessary

Differentiation destroys constant information: any two functions differing by a constant have the same derivative. So reversing differentiation cannot recover which constant the original function had, and the indefinite integral must include an arbitrary $+C$ to represent the whole family. Omitting the $+C$ is a standard scoring deduction on free-response questions asking for an indefinite integral or solving a differential equation. For definite integrals the constant cancels (previous topic), so it appears only in the indefinite case, but there it is mandatory.

Rewriting before integrating

Just as with the power rule for derivatives, the habit that prevents errors is to rewrite every term as a power of $x$ (or a standard form) before integrating. Roots become fractional powers, reciprocals become negative powers, and products or quotients are expanded or split where possible. For example $\int \frac{x^2 + 1}{x}\,dx$ should first be split into $\int \left(x + \frac{1}{x}\right)dx = \frac{x^2}{2} + \ln|x| + C$, since there is no product or quotient rule for integration to fall back on. Algebraic simplification is even more important for integration than for differentiation, because the toolkit of integration rules is smaller.

Checking an antiderivative by differentiating

Every antiderivative can be verified by differentiating it and confirming you recover the integrand. This check is fast and catches the most common slips: a missing constant factor, a sign error on $\int \sin x\,dx$, or a power-rule arithmetic mistake. After writing $\int f(x)\,dx = F(x) + C$, mentally differentiate $F$; if $F'(x) \neq f(x)$, the antiderivative is wrong. Because differentiation is more reliable than integration for most students, this reverse check is a cheap insurance policy on the no-calculator section, where there is no calculator to confirm the result. Building the habit of differentiating your answer turns integration into a self-correcting process.

The transcendental antiderivatives to memorize

Beyond the power rule, a small fixed set of antiderivatives must be known cold: $\int e^x\,dx = e^x + C$, $\int \frac{1}{x}\,dx = \ln|x| + C$, $\int \cos x\,dx = \sin x + C$, and $\int \sin x\,dx = -\cos x + C$. These come directly from reversing the corresponding derivative rules of Unit 2, so if you know the derivatives you can reconstruct the antiderivatives, watching the sign on the sine and cosine pair. The absolute value in $\ln|x|$ matters because the domain of $\frac{1}{x}$ includes negative $x$, where the antiderivative must still be defined. Fluency with this short list, combined with the power rule and linearity, covers the basic indefinite integrals the AB exam asks for before any substitution is needed.