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mathsIntegral CalculusAlgebra of Integrals

### Standard Results of Anti-Derivatives

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

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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+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
• 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
• -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
• -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
• -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
• -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 + 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
• -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
• -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
• -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
• -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
• 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
• - 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
• -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 + 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
• (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
• ln |x| + c
• 1/(x^0) + c

The answer is "ln 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

Solved Exercise Problem:

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

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