Question

Integration of x^8(1-x^6)/(1+x)^24​

Answer 1

Answer:

$x^8(1-x^6)/x^{2}(1+x)^{24}$ =  $x^7 - x^13/(1 + x)^23$

Step-by-step explanation:

From the above question,

They have given :

Integration of $x^8(1-x^6)/(1+x)^24$

Here we have to find

It is a reverse process of differentiation, where we reduce the functions into parts.

This method is used to find the summation under a vast scale. Calculation of small addition problems is an easy task which we can do manually or by using calculators as well.

The partial fraction decomposition of this expression is as follows:

$(1/24)(x-1)+(1/8)(x^2-x+1/2)+(-1/18)(x^3-x^2+x/3)+(-1/48)(x^4-x^3+x^2/2)+(-1/24)(x^5-x^4+x^3/3)+(-1/48)(x^6-x^5+x^4/2)+(-1/48)(x^7-x^6+x^5/2)+(-1/48)(x^8-x^7+x^6/2)$

We use the binomial theorem to expand the numerator and denominator.

The binomial theorem states that:

$(a + b)^n = a^n + nC1a^(n-1)b + nC2a^(n-2)b^2 + ... + nC(n-1)ab^(n-1) + b^n$

We have:

$(1 + x)^24 = 1 + 24C1x + 276C2x^2 + ... + 24C23x^23 + x^24$

and

$(1 - x^6) = 1 - 6C1x^6 + 15C2x^12 - 20C3x^18 + 15C4x^24 - 6C5x^30 + x^36$

Therefore,

$x^8(1 - x^6)/(1 + x)^24 = x^8(1 - 6C1x^6 + 15C2x^12 - 20C3x^18 + 15C4x^24 - 6C5x^30 + x^36) / (1 + 24C1x + 276C2x^2 + ... + 24C23x^23 + x^24)$

Simplifying,

$x^8(1 - x^6)/(1 + x)^24 = x^8(1 - 6C1x^6 + 15C2x^12 - 20C3x^18 + 15C4x^24 - 6C5x^30 + x^36) / (1 + 24x + 276x^2 + ... + 24x^23 + x^24)$

To simplify this expression, we can use the method of polynomial division.

We divide the numerator by the denominator, and this gives us

$x^7 - x^13/(1 + x)^23$

Therefore, the simplified expression is

$x^7 - x^13/(1 + x)^23$

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