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 often, plus enough judgment to know when a problem does not match the table.

What a table of integrals helps you do

The table is mainly a pattern-recognition tool. If the expression is already in a standard form, you can integrate directly. If it is not, the table helps you see that you probably need another method such as $u$-substitution or integration by parts.

For indefinite integrals, the goal is to find a function $F(x)$ such that

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

where $F'(x) = f(x)$. The constant $C$ is necessary because derivatives of constants are zero.

Basic table of integrals you should know

These are the entries people usually 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 just as important as any single entry:

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

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

Common table entries with $ax$ or $ax+b$

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.

The power rule exception: $\frac{1}{x}$

The power rule does not work for $n=-1$. That case becomes

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

If you try to force the power rule, the denominator becomes $n+1=0$, which is not allowed. This is the standard exception worth memorizing early.

Worked example: using the table step by step

Find

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

Each term matches a standard pattern, but not always the simplest basic one.

Use linearity to split the integral:

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

For the first term, use the power rule:

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

For the second term, use the logarithmic form with a linear inner expression. Since the denominator is $x+1$, here $a=1$, so

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

For the third term, use the cosine formula with a linear angle:

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

Combine the results:

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

This answer is valid on intervals where $x \ne -1$, because the original 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 takes you back to the original integrand, so the antiderivative is consistent.

Common mistakes when using a table of integrals

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

When a table of integrals is enough

A table of integrals is enough when the integrand is already in standard form or can be split into standard pieces with constants factored out.

It is not enough when the structure involves a product, quotient, or nested expression that does not match a table entry directly. In those cases, the table still helps because it tells you what form you are trying to reach after a rewrite or substitution.

Try a similar integral

Try

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

Before computing, name the matching formula for each term and note where a constant factor appears. Then differentiate your result to check it.