step by step solution of integration by parts of [x * arcsin(x)]?

Question

Umm, it needs to be along these lines:
(I got the first part, it's the second integration that I'm stuck on):
(1/2)*(x^2)*(arcsin(x))-(1/2)integration([x^2]/[1-x^2]^(1/2))

ok, i just need the step by step integration of:
(x^2) / (1-x^2)^(1/2)

Answer 1

In mathematics, the inverse trigonometric functions are a set of relationships closely related to the trigonometric functions. The principal inverses are listed in the following table.
Name Usual notation Definition Range of x for real result Range of usual principal value
arcsine y = arcsin(x) x = sin(y) −1 to +1 −π/2 ≤ y ≤ π/2
arccosine y = arccos(x) x = cos(y) −1 to +1 0 ≤ y ≤ π
arctangent y = arctan(x) x = tan(y) all −π/2 < y < π/2
arccosecant y = arccsc(x) x = cosec(y), y = arcsin(1/x) −∞ to −1 and 1 to ∞ −π/2 ≤ y < 0, or
0 < y ≤ π/2
arcsecant y = arcsec(x) x = sec(y), y = arccos(1/x) −∞ to −1 and 1 to ∞ 0 ≤ y < π/2, or
π/2 < y ≤ π
arccotangent y = arccot(x) x = cot(y), y = arctan(1/x) all −π/2 ≤ y < 0, or
0 < y ≤ π/2

The notations sin−1, cos−1, etc are often used for arcsin, arccos, etc, but this notation sometimes causes confusion between (e.g.) arcsin(x) and 1/sin(x).

In computer programming languages the functions arcsin, arccos, arctan, are usually called asin, acos, atan.

The inverse trigonometric function atan2(y,x) is available in many computer programming languages. It is defined as r = arctan(y/x), taking:

* if x = y = 0, then r = indefinite,
* if x > 0 and y = 0, then r = 0,
* if x < 0 and y = 0, then r = π, else
* if y < 0, then −π < r < 0,
* if y > 0, then 0 < r < π.

This function is used to find the direction from one point to another in 2-dimension Euclidean space.

Answer 2

First, the answer is:

Int(x*arcsin(x)dx) =

1/4(x*sqrt(1-x^2)+(2x^2-1)*arcsin(x))+C

To integrate x^2/sqrt(1-x^2) use the fact that:

d/dx arcsin(x) = (1-x^2)^(-0.5)

The integral answer is:

1/2*arcsin(x) - (x/2)*sqrt(1-x^2)

I think you'll see how to do it now.

Answer 3

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Answer 4

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