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integral

put into partial fraction 1 / { (x^3) + 1 }
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partial fraction of 1 / { (x^3) + 1 }

first factorise { (x^3) + 1 } using the identity for (a^3) +(b^3)
 the linear factor  (x + 1) is grouped with A / (x + 1)
the quadratic factor [ (x^2) - x  + 1 ], which cannot be factored,
is grouped with [ Bx+C ] / [ (x^2) - x  + 1]

simplify by getting rid of the denominator,
put x = (-1) on both sides to get the value of A

compare the coeffeicients of (x^2) and constant term
to get equations involving B and C and solve for them.

integration formula

partial fraction of
partial fraction of 1 / { (x^3) + 1 }
partial fraction of 1 / { (x^3) + 1 }




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problem2

derivative of ln(x + sqrt(x^2-1))
this is an example of differentiation using
function of function or chain rule
first use the derivative of ln(u)
where u =
(x + sqrt(x^2-1)
then use the derivative of x +
then the derivative of sqrt(v) where v =(x^2)-1
then finally the derivative of
(x^2)-1

*list of differentiation formula 

derivative%20of%20ln(x%20+%20sqrt(x%5E2-1))




some problems from integral calculus

* Integrate e^{x^(1/3)}dx

* evaluate integral of [e^(-x) +1] ^2

*evaluate ∫ (sin 3x)^4 dx with limits 0 to pi

* evaluate the integral of sqrt(tan  x)

* evaluate integral of { sqrt{1+(x^2)} } / {x^2}
using integration by parts


*∫ sec³x dx using integration by parts

*integral of (e^(ax) )  cosbx using integration by parts


Area using integral

* evaluate the area of an ellipse (x² /a²) + (y²/b²) =1 using integration

* find the area between y² = x and x² =y

* find the area bounded b y =x² and y = 3x

other questions and problems:


*integral calculus integration formula





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trigonometric identity and ratio of  certain standard angles

*list of differentiation formula     *integral calculus integration formula
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