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Algebra of Integrals

Algebra of Integrals

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 »  AntiDerivative Standard Results
    →  Inverse of standard results of derivatives


`int x^n dx = (x^(n+1))/(n+1) + c`

`int a dx = ax + c`

`int x^(-1) dx = ln x + c`

`int sin x dx = -cos x + c`

`int cos x dx = sinx +c`

`int sec^2 x dx = tan x + c`

`int csc^2 x dx = -cot x + c`

`int sec x tan x dx = sec x + c`

`int csc x cot x dx = -csc x + c`

`int e^x dx = e^x + c`

`int a^x dx = a^x ln a + c`

`int 1/x dx = ln x + c`

`int 1/(sqrt(1-x^2)) dx = arcsin x + c`

`int 1/(sqrt(1-x^2)) dx = -arccos x + c`

`int 1/(xsqrt(x^2-1)) dx = arcsec x + c`

`int 1/(xsqrt(x^2-1)) dx = -arc csc x + c`

`int 1/(1+x^2) dx = arctan x + c`

`int 1/(1+x^2) dx = -arc cot x + c`

Standard Results of Anti-Derivatives

plain and simple summary

nub

plain and simple summary

nub

dummy

The standard results of derivatives can be inversed to standard results of anti-derivatives.

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In this page, the standard results of derivatives are revised and the same is given in anti-derivative form.


Keep tapping on the content to continue learning.
Starting on learning "". In this page, the standard results of derivatives are revised and the same is given in anti-derivative form.

From the Fundamental theorem of calculus, the indefinite integral of a function is understood to be anti-derivative of a function.

For a given function `f(x)`, there are two possibilities to work out the result of integration `int f(x)dx = g(x)`

 • use first principle of `g(x) ``= lim_(n->oo) sum_(i=1)^n ``f((i x)/n) xx x/n`

 • use the anti-derivative property to find `g(x)` such that `d/(dx) g(x) = f(x)`.

Finding the anti-derivative is easier than solving limit of summation from the first principles.
And that is the reason, students learn differentiation ahead of integration.

Let us review the results of anti-derivatives.

Given the result `d/(dx) x^k = k x^(k-1)`
What is the anti-derivative of `x^n`?

  • `(x^(n+1))/(n+1) + c`
  • `(x^(n-1))/(n) + c`

The answer is "`(x^(n+1))/(n+1)`"

Given the result `d/(dx) ax = a`
What is the anti-derivative of `a`?

  • `0+c`
  • `ax+c`

The answer is "`ax+c`"

Given the result `d/(dx) sinx = cos x`
What is the anti-derivative of `cos x`?

  • `sin x+c`
  • `-sin x+c`

The answer is "`sin x+c`"

Given the result `d/(dx) cosx = -sin x`
What is the anti-derivative of `sin x`?

  • `cos x + c`
  • `-cos x + c`

The answer is "`-cos x + c`"

Given the result `d/(dx) tan x = sec^2 x`
What is the anti-derivative of `sec^2 x `?

  • `tan x + c`
  • `-tan x + c`

The answer is "`tan x + c`"

Given the result `d/(dx) cot x = -csc^2 x`
What is the anti-derivative of `csc^2x`?

  • `cot x + c`
  • `-cot x + c`

The answer is "`-cot x + c`"

Given the result `d/(dx) sec x = sec x tan x`
What is the anti-derivative of `sec x tan x` ?

  • `sec x + c`
  • `(sec x )/(tan x) + c`

The answer is "`sec x + c`"

Given the result `d/(dx) csc x = -csc x cot x`
What is the anti-derivative of `cscx cot x`?

  • `csc x + c`
  • `-csc x + c`

The answer is "`-csc x + c`"

Given the result `d/(dx) arcsinx = 1/(sqrt(1-x^2))`
What is the anti-derivative of `1/(sqrt(1-x^2))`?

  • `arcsin x + c`
  • `-arcsin x + c`

The answer is "`arcsin x + c`"

Given the result `d/(dx) arccos x = (-1)/(sqrt(1-x^2))`
What is the anti-derivative of `1/(sqrt(1-x^2))`?

  • `arccos x + c`
  • `-arccos x+c`

The answer is "`-arccos x + c`"

Given the result `d/(dx) arctan x = 1/(1+x^2)`
What is the anti-derivative of `1/(1+x^2)` ?

  • `arctan x + c`
  • `-arctan x + c`

The answer is "`arctan x + c`"

Given the result `d/(dx) arcsec x = 1/(|x|sqrt(x^2-1))`
What is the anti-derivative of `1/(xsqrt(x^2-1))`?

  • `arcsec x + c`
  • `sec x + c`

The answer is "`arcsec x + c`"

Given the result `d/(dx) arc csc x = (-1)/(|x|sqrt(x^2-1))`
What is the anti-derivative of `1/(x sqrt(x^2-1))`?

  • `arc csc x + c`
  • `- arc csc x + c`

The answer is "`- arc csc x + c`"

Given the result `d/(dx) arc cot x = (-1)/(1+x^2)`
What is the anti-derivative of `1/(1+x^2)`?

  • `arc cot x +c`
  • `-arc cot x +c`

The answer is "`- arc cot x + c`"

Given the result `d/(dx) e^x =e^x`
What is the anti-derivative of `e^x`?

  • `e^x + c`
  • `e^(x+1)+c`

The answer is "`e^x + c`"

Given the result `d/(dx) a^x = a^x ln a`
What is the anti-derivative of `a^x`?

  • `a^x + c`
  • `(a^x)/(ln a) + c`

The answer is "`(a^x)/(ln a) + c`"

Given the result `d/(dx) ln x = 1/x `
What is the anti-derivative of `x^(-1)`?

  • `ln |x| + c`
  • `1/(x^0) + c`

The answer is "`ln x + c`"

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Standard Results of Anti-Derivatives

`int x^n dx = (x^(n+1))/(n+1) + c`

`int a dx = ax + c`

`int x^(-1) dx = ln x + c`

`int sin x dx = -cos x + c`

`int cos x dx = sinx +c`

`int sec^2 x dx = tan x + c`

`int csc^2 x dx = -cot x + c`

`int sec x tan x dx = sec x + c`

`int csc x cot x dx = -csc x + c`

`int e^x dx = e^x + c`

`int a^x dx = a^x ln a + c`

`int 1/x dx = ln x + c`

`int 1/(sqrt(1-x^2)) dx = arcsin x + c`

`int 1/(sqrt(1-x^2)) dx = -arccos x + c`

`int 1/(xsqrt(x^2-1)) dx = arcsec x + c`

`int 1/(xsqrt(x^2-1)) dx = -arc csc x + c`

`int 1/(1+x^2) dx = arctan x + c`

`int 1/(1+x^2) dx = -arc cot x + c`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Integrate `int 1/(sqrt(1-x^2)) dx`

  • arcsin x + c
  • -arccos x + c
  • both the above

The answer is "both the above". The difference is the constant of integration `c`. Note `sin(pi/2 + theta) = cos theta`. The two expressions in the choices are equal, but for the constant.

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Progress

From the fundamental theorem of calculus, the indefinite integral of a function is understood to be anti-derivative of a function. For a given function f of x, there are two possibilities to work out the result of integration.;; Use first principles as limit of sum;; or use the anti-derivative property to find g of x such that derivative of g of x equals f of x. ;; Finding anti-derivative is easier than solving limit of summation from the first principles.;; Let us review the results of anti-derivatives.
Given the result, derivative of x power k = k multiplied x power k minus 1;; what is the anti-derivative of x power n.
1
2
The answer is "x power n + 1, over n + 1"
Given the result, derivative of a x = a;; what is the anti-derivative of a.
1
2
The answer is "a. x+c"
Given the result, derivative of sine x = cos x;; what is the anti-derivative of cos x.
1
2
The answer is "sine x + c"
Given the result, derivative of cos x = negative sine x;; what is the anti-derivative of sine x.
1
2
The answer is "negative cos x + c "
Given the result, derivative of tan x = secant squared x;; what is the anti-derivative of secant squared x.
1
2
The answer is "tan x + c"
Given the result, derivative of cot x = negative co-secant squared x ;; what is the anti-derivative of co-secant squared x.
1
2
The answer is "negative cot x + c"
Given the result, derivative of secant x, = secant x tan x;; what is the anti-derivative of secant x tan x.
1
2
The answer is "secant x + c"
Given the result, derivative of co-secant x = negative co-secant x cot x;; what is the anti-derivative of co-secant x cot x.
1
2
The answer is "negative co-secant x + c"
Given the result, derivative of inverse sine x = 1 over square root of 1 minus x squared;; what is the anti-derivative of 1 over square root of 1 minus x squared.
1
2
The answer is "inverse sine x + c"
Given the result, derivative of inverse cos x = negative 1 over square root of 1 minus x squared;; what is the anti-derivative of 1 over square root of 1 minus x squared.
1
2
The answer is "negative inverse cos x + c"
Given the result, derivative of inverse tan x = 1 over 1 + x squared;; what is the anti-derivative of 1 over 1 + x squared.
1
2
The answer is "inverse tan x + c"
Given the result, derivative of inverse secant x = 1 over mod x, square root of x squared minus 1;; what is the anti-derivative of 1 over mod x, square root of x squared minus 1.
1
2
The answer is "inverse secant x + c"
Given the result, derivative of inverse co-secant x = negative 1 over mod x, square root of x squared minus 1;; what is the anti-derivative of 1 over mod x, square root of x squared minus 1.
1
2
The answer is "negative inverse co-secant x + c"
Given the result, derivative of inverse cot = negative 1 over 1 + x squared;; what is the anti-derivative of 1 over 1 + x squared.
1
2
The answer is "negative inverse-cot x + c"
Given the result, derivative of e power x = e power x;; what is the anti-derivative of e power x.
1
2
The answer is "e power x + c"
Given the result, derivative of a power x, = a power x, natural log a;; what is the anti-derivative of a power x.
1
2
The answer is "a power x over natural log a, + c"
Given the result, derivative of natural log x = 1 over x;; what is the anti-derivative of x power negative 1.
1
2
The answer is "natural log of mod x + c"
the standard results of derivatives can be inversed to standard results of anti-derivatives.
Standard Results of Anti-Derivatives

int x^n dx = (x^(n+1))/(n+1) + c

int a dx = ax + c

int x^(-1) dx = ln x + c

int sin x dx = -cos x + c

int cos x dx = sinx +c

int sec^2 x dx = tan x + c

int csc^2 x dx = -cot x + c

int sec x tan x dx = sec x + c

int csc x cot x dx = -csc x + c

int e^x dx = e^x + c

int a^x dx = a^x ln a + c

int 1/x dx = ln x + c

int 1/(sqrt(1-x^2)) dx = arcsin x + c

int 1/(sqrt(1-x^2)) dx = -arccos x + c

int 1/(xsqrt(x^2-1)) dx = arcsec x + c

int 1/(xsqrt(x^2-1)) dx = -arc csc x + c

int 1/(1+x^2) dx = arctan x + c

int 1/(1+x^2) dx = -arc cot x + c
Integrate int 1/(sqrt(1-x^2)) dx
1
2
3
The answer is "both the above". The difference is the constant of integration c. Note: sine pi by 2 + theta = cos theta.;; The two expressions in the choices are equal, but for the constant.

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