What Are Basic Integration Formulas?
Basic integration formulas are the core antiderivatives used to reverse differentiation. To integrate a function $f(x)$ means to find a function $F(x)$ whose derivative is $f(x)$, so that $\dfrac{d}{dx}F(x) = f(x)$. Because many functions can share the same derivative (they differ only by a constant), an indefinite integral always ends with a constant of integration:
$$\int f(x),dx = F(x) + C$$
The $\int$ sign means "integrate," the $dx$ names the variable you are integrating with respect to, and $C$ is the constant of integration. Geometrically, a definite integral $\int_a^b f(x),dx$ measures the signed area between the curve and the horizontal axis from $a$ to $b$, while the indefinite integral above is the family of all antiderivatives. For the full family of results, an indefinite integrals reference collects the notation and rules in one place.
The formulas group into a few families:
The algebraic core: the power rule and its single exception, $\int \dfrac{1}{x},dx$.
The exponentials: $\int e^x,dx$ and the general base $\int a^x,dx$.
The six basic trigonometric integrals: the antiderivatives of $\sin x$, $\cos x$, $\sec^2 x$, $\csc^2 x$, $\sec x\tan x$, and $\csc x\cot x$.
The two inverse-trig-producing forms: integrals that look algebraic but whose answers are $\arcsin$ and $\arctan$.
Two structural rules let you break almost any starter problem into these pieces, so we begin there.
What Are The Constant-Multiple And Sum Rules?
Before any single formula, two rules of linearity let you split an integral into manageable parts. They are the reason you can integrate a whole polynomial term by term.
$$\int k,f(x),dx = k\int f(x),dx \qquad\text{(constant-multiple rule)}$$
$$\int \big[f(x) + g(x)\big],dx = \int f(x),dx + \int g(x),dx \qquad\text{(sum rule)}$$
A constant factor slides outside the integral sign, and the integral of a sum is the sum of the integrals. The same holds for differences, since a difference is a sum with a negative. These two rules are the reverse of the constant-multiple and sum rules for derivatives, and they hold because differentiation is itself linear.
What Is The Power Rule For Integration?
The power rule is the most-used basic integration formula. To integrate a power of $x$, raise the exponent by one and divide by the new exponent:
$$\int x^n,dx = \frac{x^{n+1}}{n+1} + C, \qquad n \neq -1$$
Check it by differentiating the answer back: $\dfrac{d}{dx}\left(\dfrac{x^{n+1}}{n+1}\right) = \dfrac{(n+1)x^{n}}{n+1} = x^n$. The result is the original integrand, so the formula is correct.
The condition $n \neq -1$ is not decoration. If $n = -1$, the formula would divide by $n + 1 = 0$, which is undefined. That single excluded case has its own antiderivative:
$$\int \frac{1}{x},dx = \ln|x| + C$$
The absolute value matters: $\dfrac{1}{x}$ is defined for negative $x$ too, and $\ln|x|$ extends the antiderivative to those values. Differentiating confirms it, $\dfrac{d}{dx}\ln|x| = \dfrac{1}{x}$ for every $x \neq 0$. This one exception is the most-missed detail in the whole topic, and it has an entry in the mistakes section below.
What Are The Exponential Integration Formulas?
The exponential function $e^x$ is its own derivative, so it is also its own antiderivative:
$$\int e^x,dx = e^x + C$$
For a general base $a > 0$ with $a \neq 1$, dividing by the natural logarithm of the base fixes the extra factor the chain rule would otherwise introduce:
$$\int a^x,dx = \frac{a^x}{\ln a} + C$$
Check the general case: $\dfrac{d}{dx}\left(\dfrac{a^x}{\ln a}\right) = \dfrac{a^x \ln a}{\ln a} = a^x$. The $\ln a$ in the denominator cancels the $\ln a$ that differentiation produces, leaving exactly $a^x$. When $a = e$, the denominator $\ln e = 1$ and the formula collapses back to $\int e^x,dx = e^x + C$.
What Are The Six Basic Trigonometric Integrals?
Reversing the derivative rules for the trigonometric functions gives six antiderivatives worth knowing on sight. The signs are the part to watch, and each is verified by differentiating the right-hand side.
Table: The six basic trigonometric integrals and the derivative that verifies each.
Integral | Result | Check by differentiating |
|---|---|---|
$\int \sin x,dx$ | $-\cos x + C$ | $\frac{d}{dx}(-\cos x) = \sin x$ |
$\int \cos x,dx$ | $\sin x + C$ | $\frac{d}{dx}(\sin x) = \cos x$ |
$\int \sec^2 x,dx$ | $\tan x + C$ | $\frac{d}{dx}(\tan x) = \sec^2 x$ |
$\int \csc^2 x,dx$ | $-\cot x + C$ | $\frac{d}{dx}(-\cot x) = \csc^2 x$ |
$\int \sec x\tan x,dx$ | $\sec x + C$ | $\frac{d}{dx}(\sec x) = \sec x\tan x$ |
$\int \csc x\cot x,dx$ | $-\csc x + C$ | $\frac{d}{dx}(-\csc x) = \csc x\cot x$ |
The pattern to hold onto: the integrals of $\sin x$, $\csc^2 x$, and $\csc x \cot x$ all pick up a minus sign, because their matching derivatives are the ones that carry a negative. The other three stay positive.
What Are The Inverse-Trig Integration Formulas?
Two integrals look purely algebraic but produce inverse trigonometric functions. They are the ones to recognise by their shape, a difference or a sum of squares under the integral.
$$\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\frac{x}{a} + C$$
$$\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\frac{x}{a} + C$$
Here $a$ is a positive constant. Verify the first: $\dfrac{d}{dx}\arcsin\dfrac{x}{a} = \dfrac{1}{a}\cdot\dfrac{1}{\sqrt{1 - (x/a)^2}} = \dfrac{1}{a}\cdot\dfrac{a}{\sqrt{a^2 - x^2}} = \dfrac{1}{\sqrt{a^2 - x^2}}$. Verify the second: $\dfrac{d}{dx}\left(\dfrac{1}{a}\arctan\dfrac{x}{a}\right) = \dfrac{1}{a}\cdot\dfrac{1/a}{1 + (x/a)^2} = \dfrac{1}{a^2 + x^2}$. Both differentiate back to their integrands, so both are correct.
The signal for these forms is the structure under the integral: a square root of a difference of squares points to $\arcsin$, and a plain sum of squares points to $\arctan$.
How Do You Use The Basic Integration Formulas?
A table of formulas only helps if you can match a problem to the right row. The method is the same every time, and it is where competitor pages stop short.
Identify the form. Decide which family the integrand belongs to, a power of $x$, an exponential, a trig function, or one of the inverse-trig shapes. Split sums into separate integrals first using the sum rule.
Pull out constants. Move any constant multiplier outside the integral with the constant-multiple rule, so the integrand matches the table cleanly.
Apply the formula. Write the matching antiderivative, taking special care with signs and with the $n \neq -1$ condition on the power rule.
Add $C$, then verify. For an indefinite integral, add the constant of integration, then differentiate your answer to confirm it returns the integrand. For a definite integral, evaluate the antiderivative at the two limits and subtract.
That last step is the habit no ranking page teaches: integration is the one operation you can always check yourself, because differentiating the answer must rebuild the question. When a product or a nested function does not match any single row, a technique such as integration by substitution rewrites it into a form that does; the wider set of approaches lives under methods of integration.
What Do Worked Examples Look Like?
Each example is fully stepped, and every antiderivative is differentiated back to verify it.
Example 1: A polynomial (power, constant-multiple, and sum rules together).
Evaluate $\displaystyle\int \left(3x^2 + 4x - 5\right),dx$.
Split the integral term by term, pull out each constant, and apply the power rule to each power of $x$ (recall that $-5 = -5x^0$, so its integral is $-5x$):
$$\int \left(3x^2 + 4x - 5\right),dx = 3\cdot\frac{x^3}{3} + 4\cdot\frac{x^2}{2} - 5x + C = x^3 + 2x^2 - 5x + C$$
Check: $\dfrac{d}{dx}\left(x^3 + 2x^2 - 5x + C\right) = 3x^2 + 4x - 5$, the original integrand.
Final answer: $\displaystyle\int \left(3x^2 + 4x - 5\right),dx = x^3 + 2x^2 - 5x + C$.
Example 2: A trig-and-exponential sum.
Evaluate $\displaystyle\int \left(2\sin x + e^x\right),dx$.
Use the sum rule, pull the $2$ outside, and apply $\int \sin x,dx = -\cos x$ and $\int e^x,dx = e^x$:
$$\int \left(2\sin x + e^x\right),dx = 2\left(-\cos x\right) + e^x + C = -2\cos x + e^x + C$$
Check: $\dfrac{d}{dx}\left(-2\cos x + e^x + C\right) = 2\sin x + e^x$, the original integrand. The sign on the $\sin x$ term is the place this example is designed to test.
Final answer: $\displaystyle\int \left(2\sin x + e^x\right),dx = -2\cos x + e^x + C$.
Example 3: A definite integral (evaluate at the limits).
Evaluate $\displaystyle\int_0^1 \left(2x + 1\right),dx$.
Integrate first, then substitute the two limits and subtract. An antiderivative of $2x + 1$ is $x^2 + x$:
$$\int_0^1 \left(2x + 1\right),dx = \Big[,x^2 + x,\Big]_0^1 = \left(1^2 + 1\right) - \left(0^2 + 0\right) = 2 - 0 = 2$$
A definite integral takes no $+C$, because the constant cancels in the subtraction. Geometrically, the answer $2$ is the area of the trapezium under the line $y = 2x + 1$ from $x = 0$ to $x = 1$, a check you can read straight off a graph.
Final answer: $\displaystyle\int_0^1 \left(2x + 1\right),dx = 2$.
What Is The Full Table Of Basic Integration Formulas?
This one table is the reference to keep beside you. Every indefinite result carries $+C$; a fuller inventory sits in the Bhanzu table of integrals and the companion list of integrals.
Table: The basic integration formulas, grouped by family.
Family | Integral | Result |
|---|---|---|
Power | $\int x^n,dx\ (n \neq -1)$ | $\dfrac{x^{n+1}}{n+1} + C$ |
Reciprocal | $\int \dfrac{1}{x},dx$ | $\ln\lvert x\rvert + C$ |
Exponential (base $e$) | $\int e^x,dx$ | $e^x + C$ |
Exponential (base $a$) | $\int a^x,dx$ | $\dfrac{a^x}{\ln a} + C$ |
Trig | $\int \sin x,dx$ | $-\cos x + C$ |
Trig | $\int \cos x,dx$ | $\sin x + C$ |
Trig | $\int \sec^2 x,dx$ | $\tan x + C$ |
Trig | $\int \csc^2 x,dx$ | $-\cot x + C$ |
Trig | $\int \sec x\tan x,dx$ | $\sec x + C$ |
Trig | $\int \csc x\cot x,dx$ | $-\csc x + C$ |
Inverse trig | $\int \dfrac{dx}{\sqrt{a^2 - x^2}}$ | $\arcsin\dfrac{x}{a} + C$ |
Inverse trig | $\int \dfrac{dx}{a^2 + x^2}$ | $\dfrac{1}{a}\arctan\dfrac{x}{a} + C$ |
Every derivative rule has a matching row here, which is why a solid table of derivatives is the fastest way to remember this table: read each derivative backwards and you have its integral.
Why Do The Basic Integration Formulas Work?
Integration is defined as the reverse of differentiation, so each formula is really a derivative rule stated in the opposite direction.
Each formula is a derivative read backwards. The power rule for integrals exists because the power rule for derivatives lowers an exponent by one and multiplies by it; undoing that means raising the exponent and dividing. The $+C$ appears because differentiation destroys constants, so reversing it cannot know which constant was there.
The exception at $n = -1$ is geometric, not arbitrary. No power of $x$ differentiates to $\dfrac{1}{x}$, but $\ln|x|$ does. The reciprocal curve still bounds a real area, so it must have an antiderivative, and that antiderivative is the natural logarithm rather than another power.
A definite integral is an accumulated area. Evaluating $F(b) - F(a)$ works because $F$ tracks the running total of area under $f$; subtracting the two endpoint values leaves exactly the area between them. This is why the constant cancels and why the geometric picture and the algebra always agree.
Seen this way, the table is not a list to memorise cold. It is the derivative rules turned around, with one honest gap at $n = -1$ that the logarithm fills.
Who Invented The Integral Sign And These Formulas?
The elongated $\int$ you write today was one person's deliberate choice, and it won out over a rival notation across a century of argument.
Two more figures shaped the basic formulas:
Jakob Bernoulli (1655–1705, Switzerland) helped fix the word "integral" for the operation, in correspondence with Leibniz over what to call the reverse of differentiation.
Isaac Newton (1643–1727, England) developed the same antiderivative rules independently through his "method of fluxions," arriving at the power rule for areas by a different road.
Where Are Basic Integration Formulas Used In The Real World?
The moment a field measures a rate and wants a total, one of these formulas does the work.
Physics and motion: integrating acceleration gives velocity, and integrating velocity gives displacement, both straight applications of the power rule to polynomial models of motion.
Engineering: the total charge delivered by a current, or the work done by a variable force, is the integral of the rate over the interval, evaluated by $F(b) - F(a)$.
Economics: total cost is the integral of marginal cost and total revenue the integral of marginal revenue, so firms recover totals from the per-unit rates they can measure.
Biology and medicine: the total drug exposure in the bloodstream is the area under a concentration-versus-time curve, an integral of an exponential-decay model built on $\int e^x,dx$.
Probability: the chance a continuous quantity lands in a range is the integral of its density over that range, which is why the inverse-trig and exponential forms appear throughout statistics.
One short table of antiderivatives quietly underlies motion, circuits, markets, medicine, and chance.
What Are The Most Common Mistakes With Basic Integration Formulas?
These four errors account for most lost marks on basic integrals, and each matches a question real students ask on r/calculus, r/learnmath, and course common-error handouts.
Forgetting the constant of integration.
Where it slips in:
A student writes $\int 2x,dx = x^2$ and stops, dropping the $+C$ on an indefinite integral.
Don't do this:
Do not leave an indefinite integral without its constant. Infinitely many functions share the same derivative, so the answer is a whole family.
The correct way:
Always close an indefinite integral with $+C$: $\int 2x,dx = x^2 + C$. Drop the constant only for a definite integral, where it cancels in the subtraction.
Using the power rule when $n = -1$.
Where it slips in:
A student applies $\int x^n,dx = \dfrac{x^{n+1}}{n+1}$ to $\int \dfrac{1}{x},dx$, producing $\dfrac{x^{0}}{0}$.
Don't do this:
Do not push $\dfrac{1}{x} = x^{-1}$ through the power rule. The formula divides by $n + 1 = 0$, which is undefined.
The correct way:
Treat $n = -1$ as the special case it is: $\int \dfrac{1}{x},dx = \ln|x| + C$. Every other real power uses the power rule.
Getting the sign wrong on $\int \sin x,dx$.
Where it slips in:
A student writes $\int \sin x,dx = \cos x + C$, copying the pattern from $\int \cos x,dx = \sin x + C$ without the minus.
Don't do this:
Do not assume the sine and cosine integrals are symmetric. Only one of them flips sign.
The correct way:
Remember $\int \sin x,dx = -\cos x + C$, and check it: $\dfrac{d}{dx}(-\cos x) = \sin x$. Differentiating the answer catches this every time.
Dropping the absolute value in $\ln|x|$.
Where it slips in:
A student writes $\int \dfrac{1}{x},dx = \ln x + C$, which is only valid for positive $x$.
Don't do this:
Do not omit the bars. Without them the antiderivative fails wherever $x$ is negative, even though $\dfrac{1}{x}$ is defined there.
The correct way:
Keep the absolute value: $\int \dfrac{1}{x},dx = \ln|x| + C$, which is a valid antiderivative for every $x \neq 0$.
Practice Problems On Basic Integration Formulas
Work each one, then check against the answer. Every answer is verified by differentiating back or by evaluating at the limits.
Evaluate $\displaystyle\int x^5,dx$.
(Answer: $\dfrac{x^6}{6} + C$.)Evaluate $\displaystyle\int \left(4x^3 - 6x + 2\right),dx$.
(Answer: $x^4 - 3x^2 + 2x + C$.)Evaluate $\displaystyle\int \left(\cos x + \sec^2 x\right),dx$.
(Answer: $\sin x + \tan x + C$.)Evaluate $\displaystyle\int 3e^x,dx$.
(Answer: $3e^x + C$.)Evaluate $\displaystyle\int_1^2 \dfrac{1}{x},dx$.
(Answer: $\big[\ln|x|\big]_1^2 = \ln 2 \approx 0.6931$.)Evaluate $\displaystyle\int \dfrac{dx}{4 + x^2}$.
(Answer: with $a = 2$, $\dfrac{1}{2}\arctan\dfrac{x}{2} + C$.)
Where Should You Go Next After Basic Integration Formulas?
The basic formulas are the launch pad for the rest of integral calculus, and several natural doors open from here.
Integration formulas. The wider catalogue, including the algebraic and logarithmic forms that build on this basic set.
Integration by substitution. The first technique for integrals that do not match a single row, by rewriting them so they do.
Definite integrals. Turn these antiderivatives into areas and totals, evaluated between two limits.
If your child is meeting integration for the first time, a live Bhanzu trainer teaches these formulas from the reverse-of-differentiation idea, checking every sign and constant, through the Bhanzu high-school math program.
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