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.

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 □

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

Why isn't there an arccos integral function?

probably because arcsin and arccos have almost identical derivatives. their derivatives are negatives of each other.

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 agree some of the things are not explained, I can't find the videos

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.

Main content

Calculus 2

Unit 1: Lesson 11

Indefinite integrals of common functions

Common integrals review

AP.CALC:

FUN‑6 (EU)

FUN‑6.C (LO)

FUN‑6.C.1 (EK)

FUN‑6.C.2 (EK)

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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Want to join the conversation?

ezaychik
Posted 6 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?
(3 votes)
Answer
- Thien D Ho
  Posted 6 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.
  (6 votes)
Rishabh Toliya
Posted 5 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)
(3 votes)
Answer
- Howard Bradley
  Posted 5 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
  
  □
  (2 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?
(2 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
  (3 votes)
Satyam Sharma
Posted 6 years ago. Direct link to Satyam Sharma's post “Why isn't there an arccos...”
Why isn't there an arccos integral function?
(2 votes)
Answer
- Navin Sawalani
  Posted 6 years ago. Direct link to Navin Sawalani's post “probably because arcsin a...”
  probably because arcsin and arccos have almost identical derivatives. their derivatives are negatives of each other.
  (2 votes)
edlarzu2
Posted 6 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.
(2 votes)
Answer
- Andrea Menozzi
  Posted 6 years ago. Direct link to Andrea Menozzi's post “I agree some of the thing...”
  I agree some of the things are not explained, I can't find the videos
  (5 votes)
Radek
Posted 6 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?
(2 votes)
Answer
- Zach Mariani
  Posted 6 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.
  (2 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?
(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
  (4 votes)
Usama Bin Mamun
Posted 2 years ago. Direct link to Usama Bin Mamun's post “Any mnemonics to remember...”
Any mnemonics to remember these? Anyone?
(2 votes)
Answer
- Road to 1 Million Energy Points!
  Posted 2 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!
  (1 vote)
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?
(2 votes)
Answer
zweiglekb
Posted 3 years ago. Direct link to zweiglekb's post “Hello, I one of the pract...”
Hello,
I one of the practice problems under u substitution: definite integrals gave the problem: ∫ 1/1+9x^2 dx. If you use u substitution you get (1/3)arctan(3x)+c. But if you refer to the integral above it states that ∫1/a^2 +x^2 dx = 1/a arctan (x/a)+c. In my example a=1 but the fraction for the solution isn't 1/1 it's 1/3. I'm confused. If you could please get back to me I'd really appreciate it. Thanks.
(2 votes)
Answer
- loumast17
  Posted 3 years ago. Direct link to loumast17's post “When you use u sub to put...”
  When you use u sub to put u in for 3x, you are also subbing in du/3 for dx, let me know if that doesn't make sense. But it should look like this now ∫1/1+u^2 du/3 Anyway, now with du/3 or in other words (1/3)du, you can bring that 1/3 out to the front, and make it look like (1/3)∫1/1+u^2 du and solve ∫ 1/1+u^2 du = arctan(u/1)+c = arctan(3x/1)+c, but you still have that 1/3 out in front. So it's (1/3)arctan(3x)+c. Again, let me know if that doesn't make sense.
  (1 vote)