Why isn't there an arccos integral function?

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

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.

I think this one is the one you are looking for: https://www.khanacademy.org/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)

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?

There are proofs out there for each trig function but it is much easier to just learn them by heart.

Why is the integral of tan(x) not listed?

If you just want to know the answer, then Sal covers it in this video: https://www.khanacademy.org/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

Any mnemonics to remember these? Anyone?

Just commit the derivatives to memory and then use the opposites to remember these!

Could someone please provide me with the proof for integral of 1/(a^2 + x^2)

1/(a² + x²) = 1/(a²(1 + x²/a²) Let x = a·tan(u) dx = a·sec²(u) du Therefore ∫1/(a² + x²) dx = ∫a·sec²(u) / a²(1 + tan²(u)) du = 1/a ∫sec²(u) / (1 + tan²(u)) du But 1 + tan²(u) = sec²(u) So ∫1/(a² + x²) dx = 1/a ∫ du = u/a + C Substituting back for u (= arctan(x/a) ) gives ∫1/(a² + x²) dx = 1/a · arctan(x/a) + C □

What is the difference between x^n dx and a^x dx? That is, why is one a polynomial and one an exponential function?

In the second function, variable x is the exponent. That is why the second one is exponential function.

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?

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).

Main content

Calculus, all content (2017 edition)

Course: Calculus, all content (2017 edition) > Unit 4

Lesson 3: Indefinite integrals of common functions

Common integrals review

Google Classroom

Review the integration rules for all the common function types.

Polynomials

integral, x, start superscript, n, end superscript, d, x, equals, start fraction, x, start superscript, n, plus, 1, end superscript, divided by, n, plus, 1, end fraction, plus, C

Radicals

\begin{aligned} \displaystyle\int\sqrt[\Large m]{x^n}\,dx&=\displaystyle\int x^{^{\Large\frac{n}{m}}}\,dx \\\\ &=\dfrac{x^{^{\Large \frac{n}{m}+1}}}{\dfrac{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

integral, sine, left parenthesis, x, right parenthesis, d, x, equals, minus, cosine, left parenthesis, x, right parenthesis, plus, C

integral, cosine, left parenthesis, x, right parenthesis, d, x, equals, sine, left parenthesis, x, right parenthesis, plus, C

integral, \sec, squared, left parenthesis, x, right parenthesis, d, x, equals, tangent, left parenthesis, x, right parenthesis, plus, C

integral, \csc, squared, left parenthesis, x, right parenthesis, d, x, equals, minus, cotangent, left parenthesis, x, right parenthesis, plus, C

integral, \sec, left parenthesis, x, right parenthesis, tangent, left parenthesis, x, right parenthesis, d, x, equals, \sec, left parenthesis, x, right parenthesis, plus, C

integral, \csc, left parenthesis, x, right parenthesis, cotangent, left parenthesis, x, right parenthesis, d, x, equals, minus, \csc, left parenthesis, x, right parenthesis, plus, 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

integral, e, start superscript, x, end superscript, d, x, equals, e, start superscript, x, end superscript, plus, C

integral, a, start superscript, x, end superscript, d, x, equals, start fraction, a, start superscript, x, end superscript, divided by, natural log, left parenthesis, a, right parenthesis, end fraction, plus, C

Integrals that are logarithmic functions

integral, start fraction, 1, divided by, x, end fraction, d, x, equals, natural log, vertical bar, x, vertical bar, plus, C

Want to learn more about integrating exponential functions and start fraction, 1, divided by, x, end fraction? Check out this video.

Want to practice integrating exponential functions and start fraction, 1, divided by, x, end fraction? Check out this exercise.

Integrals that are inverse trigonometric functions

integral, start fraction, 1, divided by, square root of, a, squared, minus, x, squared, end square root, end fraction, d, x, equals, \arcsin, left parenthesis, start fraction, x, divided by, a, end fraction, right parenthesis, plus, C

integral, start fraction, 1, divided by, a, squared, plus, x, squared, end fraction, d, x, equals, start fraction, 1, divided by, a, end fraction, \arctan, left parenthesis, start fraction, x, divided by, a, end fraction, right parenthesis, plus, C

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Satyam Sharma
Posted 7 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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(42 votes)
Answer
- David M
  Posted 7 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...”
  (44 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.
Button navigates to signup pageButton navigates to signup page
(26 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:
  
  https://www.khanacademy.org/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...”
  (12 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.
  Button navigates to signup page
  (22 votes)
Douglas Joshi
Posted 5 years ago. Direct link to Douglas Joshi's post “Why is the integral of ta...”
Why is the integral of tan(x) not listed?
Button navigates to signup pageButton navigates to signup page
(7 votes)
Answer
- Howard Bradley
  Posted 5 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:
  https://www.khanacademy.org/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
  Button navigates to signup page
  (12 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)
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Howard Bradley
  Posted 6 years ago. Direct link to Howard Bradley's post “1/(a² + x²) = 1/(a²(1 + x...”
  1/(a² + x²) = 1/(a²(1 + x²/a²)
  Let x = a·tan(u)
  dx = a·sec²(u) du
  
  Therefore ∫1/(a² + x²) dx = ∫a·sec²(u) / a²(1 + tan²(u)) du = 1/a ∫sec²(u) / (1 + tan²(u)) du
  But 1 + tan²(u) = sec²(u)
  So ∫1/(a² + x²) dx = 1/a ∫ du = u/a + C
  
  Substituting back for u (= arctan(x/a) ) gives
  ∫1/(a² + x²) dx = 1/a · arctan(x/a) + C
  
  □
  Comment on Howard Bradley's post “1/(a² + x²) = 1/(a²(1 + x...”
  (11 votes)
Usama Bin Mamun
Posted 3 years ago. Direct link to Usama Bin Mamun's post “Any mnemonics to remember...”
Any mnemonics to remember these? Anyone?
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(7 votes)
Answer
- Road to 1 Million Energy Points!
  Posted 3 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!
  Button navigates to signup page
  (4 votes)
ezaychik
Posted 7 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?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Thien D Ho
  Posted 7 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.
  Button navigates to signup page
  (6 votes)
Liang
Posted 4 months 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 4 months 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...”
  (5 votes)
Shwetashetty0220
Posted 3 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 3 years ago. Direct link to cossine's post “a^x is not a polynomial”
  a^x is not a polynomial
  Button navigates to signup page
  (4 votes)
Anny.Wu.925
Posted 5 years ago. Direct link to Anny.Wu.925's post “Under the Integrals that ...”
Under the Integrals that are inverse trigonometric functions, why there is "1/a" before arctan but not arcsin?
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(2 votes)
Answer