Common integrals review

Review the integration rules for all the common function types.

Polynomials

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

Radicals

\begin{aligned} \int \sqrt[m]{x^{n}} d x & = \int x^{^{\frac{n}{m}}} d x \\ = \frac{x^{^{\frac{n}{m} + 1}}}{\frac{n}{m} + 1} + C \end{aligned}

Want to learn more about integrating polynomials and radicals? Check out this video.

Want to practice integrating polynomials and radicals? Check out these exercises:

Indefinite integrals intro
Indefinite integrals
Indefinite integrals: advanced

Trigonometric functions

\int \sin ? (x) d x = ? \cos ? (x) + C

\int \cos ? (x) d x = \sin ? (x) + C

\int \sec^{2} ? (x) d x = \tan ? (x) + C

\int \csc^{2} ? (x) d x = ? \cot ? (x) + C

\int \sec ? (x) \tan ? (x) d x = \sec ? (x) + C

\int \csc ? (x) \cot ? (x) d x = ? \csc ? (x) + C

Want to learn more about integrating trigonometric functions? Check out this video.

Want to practice integrating trigonometric functions? Check out these exercises:

Indefinite integrals: sin & cos
Integrating trig functions

Exponential functions

\int e^{x} d x = e^{x} + C

\int a^{x} d x = \frac{a^{x}}{\ln ? (a)} + C

Integrals that are logarithmic functions

\int \frac{1}{x} d x = \ln ? | x | + C

Want to learn more about integrating exponential functions and $\frac{1}{x}$ ?? Check out this video.

Want to practice integrating exponential functions and $\frac{1}{x}$ ?? Check out this exercise.

Integrals that are inverse trigonometric functions

\int \frac{1}{\sqrt{a^{2} ? x^{2}}} d x = \arcsin ? (\frac{x}{a}) + C

\int \frac{1}{a^{2} + x^{2}} d x = \frac{1}{a} \arctan ? (\frac{x}{a}) + C

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Satyam Sharma
Posted 8 years ago. Direct link to Satyam Sharma's post “Why isn't there an arccos...”
Why isn't there an arccos integral function?
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(44 votes)
Answer
- David M
  Posted 8 years ago. Direct link to David M's post “Basically, because the al...”
  Basically, because the algebra doesn't work out nicely. There are plenty of derivatives of trig functions that exist, but there are only a few that result in a non-trig-function-involving equation.
  For example, the derivative of arcsin(x/a)+c = 1/sqrt(a^2-x^2), doesn't involve any trig functions in it's derivative. If we reverse this process on 1/sqrt(a^2-x^2) (find the indefinite integral) we get arcsin(x/a)+C, so we went from an equation with no trig functions to an equation with trig functions.
  
  There aren't many other equations that work out this nicely.
  https://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functions#Integrands_involving_only_cosine
  Comment on David M's post “Basically, because the al...”
  (53 votes)
edlarzu2
Posted 7 years ago. Direct link to edlarzu2's post “Where is the video for th...”
Where is the video for the second function under "Exponential Functions" (the integral of a^x)? Where are the videos for the whole section of "Integrals that are inverse trigonometric functions" (the integral of 1/sqrt[(a^2)-(x^2)] and the integral of 1/sqrt[(a^2)+(x^2)]? I can't find these three functions mentioned anywhere in the videos.
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(28 votes)
Answer
- Lucas
  Posted 5 years ago. Direct link to Lucas's post “I think this one is the o...”
  I think this one is the one you are looking for:
  
  /math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1b/v/exponential-functions-differentiation-intro
  
  (I know this is a bit late, but perhaps other people can get use out of this)
  Comment on Lucas's post “I think this one is the o...”
  (16 votes)
Radek
Posted 7 years ago. Direct link to Radek's post “All these integrals of tr...”
All these integrals of trigonometric functions are really confusing for me. Do I have to just learn them by heart? Or is there some section I missed, where they are explained more intuitively?
Button navigates to signup pageComment on Radek's post “All these integrals of tr...”
(8 votes)
Answer
- Zach Mariani
  Posted 7 years ago. Direct link to Zach Mariani's post “There are proofs out ther...”
  There are proofs out there for each trig function but it is much easier to just learn them by heart.
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  (27 votes)
Douglas Joshi
Posted 6 years ago. Direct link to Douglas Joshi's post “Why is the integral of ta...”
Why is the integral of tan(x) not listed?
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(8 votes)
Answer
- Howard Bradley
  Posted 6 years ago. Direct link to Howard Bradley's post “If you just want to know ...”
  If you just want to know the answer, then Sal covers it in this video:
  /math/calculus-home/integration-techniques-calc/reverse-chain-rule-calc/v/integral-of-tan-x
  
  If you feel it's an omission that needs correction (I'd agree) then you might like to raise a "feature request":
  https://khanacademy.zendesk.com/hc/en-us/community/topics/200136634-Feature-Requests
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  (14 votes)
Usama Bin Mamun
Posted 4 years ago. Direct link to Usama Bin Mamun's post “Any mnemonics to remember...”
Any mnemonics to remember these? Anyone?
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(8 votes)
Answer
- Road to 1 Million Energy Points!
  Posted 4 years ago. Direct link to Road to 1 Million Energy Points!'s post “Just commit the derivativ...”
  Just commit the derivatives to memory and then use the opposites to remember these!
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  (5 votes)
Rishabh Toliya
Posted 6 years ago. Direct link to Rishabh Toliya's post “Could someone please prov...”
Could someone please provide me with the proof for
integral of 1/(a^2 + x^2)
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(4 votes)
Answer
- Howard Bradley
  Posted 6 years ago. Direct link to Howard Bradley's post “1/(a? + x?) = 1/(a?(1 + x...”
  1/(a? + x?) = 1/(a?(1 + x?/a?)
  Let x = a·tan(u)
  dx = a·sec?(u) du
  
  Therefore ∫1/(a? + x?) dx = ∫a·sec?(u) / a?(1 + tan?(u)) du = 1/a ∫sec?(u) / (1 + tan?(u)) du
  But 1 + tan?(u) = sec?(u)
  So ∫1/(a? + x?) dx = 1/a ∫ du = u/a + C
  
  Substituting back for u (= arctan(x/a) ) gives
  ∫1/(a? + x?) dx = 1/a · arctan(x/a) + C
  
  □
  Comment on Howard Bradley's post “1/(a? + x?) = 1/(a?(1 + x...”
  (12 votes)
ezaychik
Posted 8 years ago. Direct link to ezaychik's post “What is the difference be...”
What is the difference between x^n dx and a^x dx? That is, why is one a polynomial and one an exponential function?
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(3 votes)
Answer
- Thien D Ho
  Posted 8 years ago. Direct link to Thien D Ho's post “In the second function, v...”
  In the second function, variable x is the exponent. That is why the second one is exponential function.
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  (6 votes)
Liang
Posted a year ago. Direct link to Liang's post “at Integrals that are inv...”
at Integrals that are inverse trigonometric functions above:

I took d/dx (arcsin x/a)=1/a *1/√1-x^2 , which does not equal 1/√a^2-x^2.

since d/dx arcsin x=1/√1-x^2, what did I do wrong?
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(2 votes)
Answer
- Venkata
  Posted a year ago. Direct link to Venkata's post “There's a small error you...”
  There's a small error you made. You were right on using the chain rule by multiplying the 1/a, but observe that you need to take the derivative of arcsin(x/a) w.r.t (x/a). You took it w.r.t x. So, you'd get 1/(√1-(x/a)^2) * 1/a. With some simplification, this turns into 1/(√a^2-x^2).
  Comment on Venkata's post “There's a small error you...”
  (7 votes)
Shwetashetty0220
Posted 4 years ago. Direct link to Shwetashetty0220's post “in common integral review...”
in common integral review under exponeential functions how integration of ax is ax/ln{a} shoudnt it be like the polynomial example?
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(1 vote)
Answer
- cossine
  Posted 4 years ago. Direct link to cossine's post “a^x is not a polynomial”
  a^x is not a polynomial
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  (5 votes)
hasti.g1384
Posted 2 years ago. Direct link to hasti.g1384's post “something confused me, fo...”
something confused me, for calculating definite integrals we must have a graph or details which we can calculate the area from them; is it right?

But for calculating indefinite integrals, using such simple rules is enough.
Am I correct?
Button navigates to signup pageComment on hasti.g1384's post “something confused me, fo...”
(2 votes)
Answer
- JPOgle ?
  Posted a year ago. Direct link to JPOgle ?'s post “For calculating definite ...”
  For calculating definite integrals, you can use a graph, but you can also use the second part of the fundamental theorem of calculus. For indefinite integrals, the simple rules are enough for now, but it gets way more complicated later.
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  (3 votes)