Integration Formula Sheet

An integration formula sheet is a quick-reference guide for common indefinite integral results. When you solve a problem, the decisive question is not "how many formulas have I memorized?" but "can this integrand be matched directly to a standard form?" If the expression is a power function, $1/x$, an exponential, or a common trigonometric function, the formulas apply directly. If it is a product, a composite, or a complex fraction, you usually need substitution, integration by parts, or simplification first.

The Formulas And What Each Symbol Means

Each row pairs a recognizable integrand with its antiderivative; $C$ is the constant of integration, added because every antiderivative is unique only up to a constant.

Type	Formula	Conditions or Tips
Power Function	\int x^n\,dx = \frac\{x^\{n+1\}}\{n+1\} + C	Only holds when $n \ne -1$
Logarithmic Type	$\int \frac{1}{x},dx = \ln	x
Exponential Function	$\int e^x\,dx = e^x + C$	Base is the natural constant $e$
General Exponential	$\int a^x\,dx = \frac\{a^x\}\{\ln a\} + C$	Requires $a > 0$ and $a \ne 1$
Sine Function	$\int \sin x\,dx = -\cos x + C$	Easy to miss the negative sign
Cosine Function	$\int \cos x\,dx = \sin x + C$	Sign is opposite to the one above
Secant Squared	$\int \sec^2 x\,dx = \tan x + C$	Common in anti-derivative problems
Arctangent Type	$\int \frac\{1\}\{1+x^2\}\,dx = \arctan x + C$	Denominator must be in the standard form of $1+x^2$

The linearity property ties the table together:

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

So sums, differences, and constant multiples can be handled separately, but products cannot be split apart:

$\int f(x)g(x)\,dx \ne \left(\int f(x)\,dx\right)\left(\int g(x)\,dx\right)$

Why These Formulas Hold: Reverse Differentiation

Every entry is just a derivative read backwards, which is why differentiating the result always returns the integrand. Take the power rule: differentiating $\frac{x^{n+1}}{n+1}$ gives $\frac{(n+1)x^n}{n+1} = x^n$, recovering the original. This also explains the single exception. The most critical condition in the power rule is $n \ne -1$, because at $n = -1$ you would divide by $n+1 = 0$. There $x^n = x^{-1} = \frac{1}{x}$, whose antiderivative is logarithmic, not a power:

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

This is why writing $\int x^{-1}\,dx$ as $\frac{x^0}{0}$ fails: the denominator becomes $0$, signaling the power rule no longer applies. Knowing the formulas are reversed derivatives gives you a built-in check for every problem.

Worked Example, Step By Step

Find

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

The integrand is a sum of three terms, each matching a row in the sheet, so integrate them separately. First term, power rule:

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

Second term, sine formula:

$\int -4\sin x\,dx = 4\cos x$

Third term, arctangent formula:

$\int \frac{5}{1+x^2}\,dx = 5\arctan x$

Combine:

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

Now use the reverse-differentiation check:

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

We return to the original expression, so the result is correct.

Try It Yourself, Then Check

Solve

$\int \left(6x - 2\cos x + \frac{3}{1+x^2}\right)\,dx$

Then verify just three things: does each term truly match a formula, did you include $+C$, and does differentiating the result return the original integrand? Afterward, try a problem that needs substitution or integration by parts to feel where the formula sheet ends and other techniques begin.

Calculation Pitfalls

1. Forgetting the $+C$. Every indefinite integral needs the constant of integration; only definite integrals give a single number after substituting limits.
2. Treating $x^{-1}$ as a power function. $\int x^{-1}\,dx = \ln|x| + C$; the power rule does not apply.
3. Swapping trigonometric signs. $\int \sin x\,dx = -\cos x + C$, whereas $\int \cos x\,dx = \sin x + C$; they look alike but the signs differ.
4. Forcing formulas onto products. A product like $x e^x$ or $x\cos x$ usually needs integration by parts, and an inner function like $\cos(3x+1)$ usually needs substitution. Check the structure before reaching for the table.

When To Use The Sheet

The formula sheet shines for quickly finding antiderivatives of indefinite integrals, and it underpins advanced methods too: before substitution you must recognize the target form, and after integration by parts you return to these basics to finish. If the problem is already in standard pattern, the table is highly efficient; if it is not yet there, do not rush to apply formulas, or you may head in the wrong direction.