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 cosx is sinx because the derivative of sinx is cosx. The integral of 1/(1+x2) is arctanx because the derivative of arctanx is 1/(1+x2).
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.
Why Memorize Special Integrals?
Integration techniques transform unfamiliar integrals into recognizable ones. These standard forms are the targets.
Knowing them provides:
Each formula inverts a differentiation rule. Mastery of derivatives implies knowledge of these integrals.
Power Functions
The power rule for integration:
∫xndx=n+1xn+1+C(n=−1)
The exponent increases by one; divide by the new exponent.
The exception n=−1 is critical:
∫x−1dx=∫x1dx=ln∣x∣+C
The absolute value ensures validity for negative x. For x>0, the antiderivative is lnx; for x<0, it is ln(−x). Both cases combine into ln∣x∣.
Exponential Functions
The exponential function is its own antiderivative:
∫exdx=ex+C
For other bases:
∫axdx=lnaax+C(a>0,a=1)
The factor 1/lna compensates for the chain rule in (ax)′=axlna.
With a linear argument:
∫ekxdx=kekx+C
Trigonometric Functions
Basic trigonometric integrals:
∫sinxdx=−cosx+C
∫cosxdx=sinx+C
∫sec2xdx=tanx+C
∫csc2xdx=−cotx+C
∫secxtanxdx=secx+C
∫cscxcotxdx=−cscx+C
Each reverses a standard derivative. The negative signs in sinx and csc2x integrals reflect the negatives in (cosx)′=−sinx and (cotx)′=−csc2x.
Inverse Trigonometric Forms
These integrals produce inverse trigonometric functions:
∫1+x21dx=arctanx+C
∫1−x21dx=arcsinx+C
∫xx2−11dx=arcsec∣x∣+C
More generally, with constant a>0:
∫a2+x21dx=a1arctanax+C
∫a2−x21dx=arcsinax+C
Logarithmic Integrals
The logarithm itself requires integration by parts:
∫lnxdx=xlnx−x+C
A crucial pattern recognizes when integrands have the form f′(x)/f(x):
∫f(x)f′(x)dx=ln∣f(x)∣+C
This pattern appears frequently in disguise.
Example:
∫tanxdx=∫cosxsinxdx=−ln∣cosx∣+C
Here f(x)=cosx and f′(x)=−sinx, so the integrand is −f′(x)/f(x).
Integrals Leading to Logarithms
Several standard integrals produce logarithms:
∫tanxdx=−ln∣cosx∣+C=ln∣secx∣+C
∫cotxdx=ln∣sinx∣+C
∫secxdx=ln∣secx+tanx∣+C
∫cscxdx=−ln∣cscx+cotx∣+C=ln∣cscx−cotx∣+C
The secx and cscx integrals are less obvious—they require multiplying by clever forms of 1 to create the f′/f pattern.