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Special Interals



Why Memorize Special Integrals?
Power Functions
Exponential Functions
Trigonometric Functions
Inverse Trigonometric Forms
Logarithmic Integrals
Integrals Leading to Logarithms
Master Reference: All Special Integrals at a Glance
Special Integrals FAQ



Formulas Worth Knowing


Certain integrals appear so frequently that recognizing them on sight saves considerable effort. These standard forms serve as endpoints—the targets that techniques aim to reach.

Each formula below reverses a known derivative. The integral of cos⁡x\cos x is sin⁡x\sin x because the derivative of sin⁡x\sin x is cos⁡x\cos x. The integral of 1/(1+x2)1/(1 + x^2) is arctan⁡x\arctan x because the derivative of arctan⁡x\arctan x is 1/(1+x2)1/(1 + x^2).

Memorizing these forms accelerates computation and provides reference points for more complex integrals. When a technique transforms an integral into one of these patterns, the work is essentially done.

Key Terms

Antiderivative— each formula reverses a known derivative
Indefinite Integral— the notation ∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C
Integrand— recognizing the integrand's form selects the formula

See All Calculus Definitions →


Power Functions


The power rule for integration:

Power Rule (Integrals)
∫xn dx=xn+1n+1+Cn≠−1\int x^n\, dx = \frac{x^{n+1}}{n+1} + C \quad n \neq -1
Learn more about this formula: Power Rule (Integrals) →

The exponent increases by one; divide by the new exponent.

The exception n=−1n = -1 is critical:

Reciprocal Antiderivative
∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\, dx = \ln|x| + C
Learn more about this formula: Reciprocal Antiderivative →


The absolute value ensures validity for negative xx. For x>0x > 0, the antiderivative is ln⁡x\ln x; for x<0x < 0, it is ln⁡(−x)\ln(-x). Both cases combine into ln⁡∣x∣\ln|x|.

Exponential Functions


The exponential function is its own antiderivative:

Exponential Antiderivative
∫ex dx=ex+C\int e^x\, dx = e^x + C
Learn more about this formula: Exponential Antiderivative →

For other bases:

General Exponential Antiderivative
∫ax dx=axln⁡a+Ca>0, a≠1\int a^x\, dx = \frac{a^x}{\ln a} + C \quad a > 0, \, a \neq 1
Learn more about this formula: General Exponential Antiderivative →


The factor 1/ln⁡a1/\ln a compensates for the chain rule in (ax)′=axln⁡a(a^x)' = a^x \ln a.

With a linear argument:

∫ekx dx=ekxk+C\int e^{kx}\, dx = \frac{e^{kx}}{k} + C


Trigonometric Functions


Basic trigonometric integrals:

Antiderivative of Sine
∫sin⁡x dx=−cos⁡x+C\int \sin x\, dx = -\cos x + C
Learn more about this formula: Antiderivative of Sine →

Antiderivative of Cosine
∫cos⁡x dx=sin⁡x+C\int \cos x\, dx = \sin x + C
Learn more about this formula: Antiderivative of Cosine →

Antiderivative of Sec Squared
∫sec⁡2x dx=tan⁡x+C\int \sec^2 x\, dx = \tan x + C
Learn more about this formula: Antiderivative of Sec Squared →

Antiderivative of Csc Squared
∫csc⁡2x dx=−cot⁡x+C\int \csc^2 x\, dx = -\cot x + C
Learn more about this formula: Antiderivative of Csc Squared →

Antiderivative of Sec Tan
∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec x \tan x\, dx = \sec x + C
Learn more about this formula: Antiderivative of Sec Tan →

Antiderivative of Csc Cot
∫csc⁡xcot⁡x dx=−csc⁡x+C\int \csc x \cot x\, dx = -\csc x + C
Learn more about this formula: Antiderivative of Csc Cot →


Each reverses a standard derivative. The negative signs in sin⁡x\sin x and csc⁡2x\csc^2 x integrals reflect the negatives in (cos⁡x)′=−sin⁡x(\cos x)' = -\sin x and (cot⁡x)′=−csc⁡2x(\cot x)' = -\csc^2 x.
Integrand Antiderivative Reverses derivative of
sin x −cos x + C cos x (which gives −sin x)
cos x sin x + C sin x
sec² x tan x + C tan x
csc² x −cot x + C cot x (which gives −csc² x)
sec x · tan x sec x + C sec x
csc x · cot x −csc x + C csc x (which gives −csc x cot x)

Inverse Trigonometric Forms


These integrals produce inverse trigonometric functions:

Arctangent Form
∫11+x2 dx=arctan⁡x+C\int \frac{1}{1 + x^2}\, dx = \arctan x + C
Learn more about this formula: Arctangent Form →

Arcsine Form
∫11−x2 dx=arcsin⁡x+C\int \frac{1}{\sqrt{1 - x^2}}\, dx = \arcsin x + C
Learn more about this formula: Arcsine Form →

Arcsecant Form
∫1xx2−1 dx=arcsec⁡∣x∣+C\int \frac{1}{x\sqrt{x^2 - 1}}\, dx = \operatorname{arcsec}|x| + C
Learn more about this formula: Arcsecant Form →


More generally, with constant a>0a > 0:

∫1a2+x2 dx=1aarctan⁡xa+C\int \frac{1}{a^2 + x^2}\, dx = \frac{1}{a}\arctan\frac{x}{a} + C


∫1a2−x2 dx=arcsin⁡xa+C\int \frac{1}{\sqrt{a^2 - x^2}}\, dx = \arcsin\frac{x}{a} + C

Integrand Antiderivative Note
1 / (1 + x²) arctan x + C base form (a = 1)
1 / √(1 − x²) arcsin x + C base form (a = 1); valid on |x| < 1
1 / (x · √(x² − 1)) arcsec |x| + C absolute value handles both x > 1 and x < −1
1 / (a² + x²) (1/a) arctan(x/a) + C generalization with a > 0
1 / √(a² − x²) arcsin(x/a) + C generalization with a > 0; valid on |x| < a

Logarithmic Integrals


The logarithm itself requires integration by parts:

Antiderivative of Natural Log
∫ln⁡x dx=xln⁡x−x+C\int \ln x\, dx = x\ln x - x + C
Learn more about this formula: Antiderivative of Natural Log →

A crucial pattern recognizes when integrands have the form f′(x)/f(x)f'(x)/f(x):

Logarithmic Derivative Pattern
∫f′(x)f(x) dx=ln⁡∣f(x)∣+C\int \frac{f'(x)}{f(x)}\, dx = \ln|f(x)| + C
Learn more about this formula: Logarithmic Derivative Pattern →


This pattern appears frequently in disguise.

Example:

∫tan⁡x dx=∫sin⁡xcos⁡x dx=−ln⁡∣cos⁡x∣+C\int \tan x\, dx = \int \frac{\sin x}{\cos x}\, dx = -\ln|\cos x| + C


Here f(x)=cos⁡xf(x) = \cos x and f′(x)=−sin⁡xf'(x) = -\sin x, so the integrand is −f′(x)/f(x)-f'(x)/f(x).

Integrals Leading to Logarithms


Several standard integrals produce logarithms:

Antiderivative of Tangent
∫tan⁡x dx=−ln⁡∣cos⁡x∣+C=ln⁡∣sec⁡x∣+C\int \tan x\, dx = -\ln|\cos x| + C = \ln|\sec x| + C
Learn more about this formula: Antiderivative of Tangent →

Antiderivative of Cotangent
∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x\, dx = \ln|\sin x| + C
Learn more about this formula: Antiderivative of Cotangent →

Antiderivative of Secant
∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+C\int \sec x\, dx = \ln|\sec x + \tan x| + C
Learn more about this formula: Antiderivative of Secant →

Antiderivative of Cosecant
∫csc⁡x dx=−ln⁡∣csc⁡x+cot⁡x∣+C=ln⁡∣csc⁡x−cot⁡x∣+C\int \csc x\, dx = -\ln|\csc x + \cot x| + C = \ln|\csc x - \cot x| + C
Learn more about this formula: Antiderivative of Cosecant →


The sec⁡x\sec x and csc⁡x\csc x integrals are less obvious—they require multiplying by clever forms of 11 to create the f′/ff'/f pattern.

Master Reference: All Special Integrals at a Glance


The formulas covered in the sections above all share the same role: standard endpoints that techniques aim to reach. The master reference below collects every formula on this page in one place, grouped by family so that scanning by integrand form returns the matching antiderivative directly. Use it as a lookup card while working integrals — the section above each row supplies the derivation and the context.
Family Integrand Antiderivative (+ C)
Power xn  (n ≠ −1) xn+1 / (n + 1)
Power 1 / x ln |x|
Exponential ex ex
Exponential ax  (a > 0, a ≠ 1) ax / ln a
Exponential ekx ekx / k
Trigonometric sin x −cos x
Trigonometric cos x sin x
Trigonometric sec² x tan x
Trigonometric csc² x −cot x
Trigonometric sec x · tan x sec x
Trigonometric csc x · cot x −csc x
Inverse trig 1 / (1 + x²) arctan x
Inverse trig 1 / √(1 − x²) arcsin x
Inverse trig 1 / (x · √(x² − 1)) arcsec |x|
Logarithmic ln x x ln x − x
Logarithmic pattern f'(x) / f(x) ln |f(x)|
Trig → log tan x −ln |cos x|  =  ln |sec x|
Trig → log cot x ln |sin x|
Trig → log sec x ln |sec x + tan x|
Trig → log csc x −ln |csc x + cot x|  =  ln |csc x − cot x|

Special Integrals FAQ

What is the power rule for integration?

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∫xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for every n except −1. Raise the exponent by one, then divide by the new exponent — the exact reverse of the derivative power rule, which lowers the exponent and multiplies by the old one. The exclusion of n = −1 is not a technicality: dividing by n + 1 would divide by zero.Read more →

Why is ∫(1/x) dx = ln|x| instead of ln x?

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Because 1/x is defined for negative x but ln x is not. For x > 0 the antiderivative is ln x; for x < 0 it is ln(−x), whose derivative is also 1/x by the chain rule. The absolute value packs both branches into one formula. Strictly, each branch carries its own constant, since the domain splits at zero.Read more →

What is the integral of a^x?

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∫aˣ dx = aˣ/ln a + C, for a > 0 and a ≠ 1. The 1/ln a compensates for the chain rule: differentiating aˣ produces aˣ ln a, so the antiderivative has to divide that factor back out. When a = e the factor is ln e = 1, which is why ∫eˣ dx = eˣ + C carries no coefficient.Read more →

Why is ∫sin x dx = −cos x?

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Because the derivative that produces sin x carries a minus sign. Since (cos x)′ = −sin x, reversing it means the antiderivative of sin x is −cos x, not cos x. Check by differentiating: (−cos x)′ = sin x. The same inversion explains ∫csc²x dx = −cot x, since (cot x)′ = −csc²x.Read more →

How do you spot an integral that gives a logarithm?

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Look for the pattern f′(x)/f(x) — a function in the denominator with its own derivative on top. Any integrand of that shape integrates to ln|f(x)| + C. It shows up in disguise constantly: ∫tan x dx is ∫sin x / cos x dx, which is −f′/f with f = cos x, giving −ln|cos x| + C.Read more →

What is the integral of sec x?

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∫sec x dx = ln|sec x + tan x| + C. It is the least obvious of the standard trigonometric integrals, because you get there by multiplying by a clever form of 1: sec x · (sec x + tan x)/(sec x + tan x). The numerator then becomes the derivative of the denominator, turning it into the f′/f pattern.Read more →