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## Calculus, all content (2017 edition)

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

Lesson 3: Indefinite integrals of common functions- Indefinite integrals of sin(x), cos(x), and eˣ
- Indefinite integral of 1/x
- Indefinite integrals: eˣ & 1/x
- Particular solutions to differential equations: rational function
- Particular solutions to differential equations: exponential function
- Particular solutions to differential equations
- Indefinite integrals: sin & cos
- Integrating trig functions
- Antiderivatives and indefinite integrals review
- Common integrals review

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# Common integrals review

Review the integration rules for all the common function types.

## Polynomials

## Radicals

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

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

## Trigonometric functions

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

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

## Exponential functions

## Integrals that are logarithmic functions

*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

## Want to join the conversation?

- Why isn't there an arccos integral function?(42 votes)
- 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(44 votes)

- 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.(26 votes)
- 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)(12 votes)

- 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?(8 votes)
- There are proofs out there for each trig function but it is much easier to just learn them by heart.(22 votes)

- Why is the integral of tan(x) not listed?(7 votes)
- 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(12 votes)

- Could someone please provide me with the proof for

integral of 1/(a^2 + x^2)(4 votes)- 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

□(11 votes)

- Any mnemonics to remember these? Anyone?(7 votes)
- Just commit the derivatives to memory and then use the opposites to remember these!(4 votes)

- 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)
- In the second function, variable x is the exponent. That is why the second one is exponential function.(6 votes)

- 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?(2 votes)- 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).(5 votes)

- in common integral review under exponeential functions how integration of ax is ax/ln{a} shoudnt it be like the polynomial example?(1 vote)
- a^x is not a polynomial(4 votes)

- Under the Integrals that are inverse trigonometric functions, why there is "1/a" before arctan but not arcsin?(2 votes)