Integration Formulas — Complete Table of Integrals

A table of integrals is a list of standard antiderivatives. Use it when the integrand already matches a known pattern such as $x^n$, $\frac{1}{x}$, $e^x$, or a basic trig function. No finite table is literally complete for every possible integral; in practice a "complete table of integrals" means the standard formulas students use most, plus the judgment to know when a problem does not match the table.

The Table And What The Symbols Mean

The table is a pattern-recognition tool. For indefinite integrals the goal is a function $F(x)$ with

$$\int f(x)\,dx = F(x) + C$$

where $F'(x) = f(x)$. The constant $C$ is necessary because derivatives of constants are zero. These are the core entries people mean when they ask for an integrals table:

Type	Formula	Condition
Power rule	\int x^n\,dx = \frac\{x^\{n+1\}}\{n+1\} + C	$n \ne -1$
Logarithmic case	$\int \frac{1}{x},dx = \ln	x
Natural exponential	$\int e^x\,dx = e^x + C$	none
Exponential base $a$	$\int a^x\,dx = \frac\{a^x\}\{\ln a\} + C$	$a > 0$, $a \ne 1$
Sine	$\int \sin x\,dx = -\cos x + C$	none
Cosine	$\int \cos x\,dx = \sin x + C$	none
Secant squared	$\int \sec^2 x\,dx = \tan x + C$	where defined
Cosecant squared	$\int \csc^2 x\,dx = -\cot x + C$	where defined
Reciprocal quadratic	$\int \frac\{1\}\{1+x^2\}\,dx = \arctan x + C$	none
Inverse sine form	\int \frac\{1\}\{\sqrt\{1-x^2\}}\,dx = \arcsin x + C	valid on intervals with $

The linearity rule is as important as any single entry:

$$\int \left(af(x) + bg(x)\right)\,dx = a\int f(x)\,dx + b\int g(x)\,dx$$

It lets you split sums and pull out constants, but it does not let you split a product in general.

Why The Linear-Inner-Term Rows Work

A basic formula often reappears with $ax$ or $ax+b$ inside it. If $a \ne 0$, these are common direct results:

Type	Formula	Condition
Power with linear inner term	\int (ax+b)^n\,dx = \frac\{(ax+b)^\{n+1\}}\{a(n+1)\} + C	$a \ne 0$, $n \ne -1$
Log form with linear inner term	$\int \frac{1}{ax+b},dx = \frac{1}{a}\ln	ax+b
Exponential with linear exponent	$\int e^\{ax\}\,dx = \frac\{1\}\{a\}e^\{ax\} + C$	$a \ne 0$
Sine with linear angle	$\int \sin(ax)\,dx = -\frac\{1\}\{a\}\cos(ax) + C$	$a \ne 0$
Cosine with linear angle	$\int \cos(ax)\,dx = \frac\{1\}\{a\}\sin(ax) + C$	$a \ne 0$

These are not new ideas. They are the same standard antiderivatives with a constant-factor adjustment: differentiating the inner $ax+b$ produces a factor of $a$ by the chain rule, so dividing by $a$ when you integrate cancels it back out. That is exactly why each row carries a $\frac{1}{a}$.

The same chain-rule logic exposes the one exception. The power rule does not work for $n = -1$, because the antiderivative becomes

$$\int x^{-1}\,dx = \int \frac{1}{x}\,dx = \ln|x| + C$$

Forcing the power rule makes the denominator $n+1 = 0$, which is not allowed, so this case is the standard exception worth memorizing early.

Worked Example, Step By Step

Find

$$\int \left(3x^2 - \frac{4}{x+1} + 5\cos(2x)\right)\,dx$$

Each term matches a standard pattern, though not always the simplest one. Use linearity to split it:

$$\int 3x^2\,dx - 4\int \frac{1}{x+1}\,dx + 5\int \cos(2x)\,dx$$

First term, power rule:

$$\int 3x^2\,dx = x^3$$

Second term, log form with a linear inner expression, where $a = 1$:

$$-4\int \frac{1}{x+1}\,dx = -4\ln|x+1|$$

Third term, cosine with a linear angle, where $a = 2$ contributes the $\frac{1}{2}$:

$$5\int \cos(2x)\,dx = \frac{5}{2}\sin(2x)$$

Combine:

$$\int \left(3x^2 - \frac{4}{x+1} + 5\cos(2x)\right)\,dx = x^3 - 4\ln|x+1| + \frac{5}{2}\sin(2x) + C$$

This is valid where $x \ne -1$, since the integrand is undefined at $x=-1$. The fastest check is differentiation:

$$\frac{d}{dx}\left(x^3 - 4\ln|x+1| + \frac{5}{2}\sin(2x)\right) = 3x^2 - \frac{4}{x+1} + 5\cos(2x)$$

That returns the original integrand, so the antiderivative is consistent.

Try It Yourself

Find

$$\int \left(4x^3 + \frac{6}{x-2} - 3e^{5x}\right)\,dx$$

Before computing, name the matching formula for each term and note where a constant factor (the $\frac{1}{a}$) appears. Then differentiate your result to check it.

Calculation Pitfalls

• Matching the wrong pattern. A product like $xe^x$ or a composition like $\cos(x^2)$ is usually not a direct lookup.
• Forgetting the scaling factor. $\int \cos(2x)\,dx = \frac{1}{2}\sin(2x) + C$, not $\sin(2x) + C$.
• Using the power rule on $\frac{1}{x}$. That case is logarithmic, not another power.
• Dropping the $+C$. An indefinite integral is a family of antiderivatives, not one single function.

When The Table Is Enough

A table of integrals is enough when the integrand is already in standard form or splits into standard pieces with constants factored out. It is not enough when a product, quotient, or nested expression does not match an entry directly. Even then the table helps, because it tells you what form you are aiming for after a rewrite or substitution.