What is the integral of (x^2)/((x-1)^{100)} ?

The integral of (x^2)/((x-1)^{100)} is 1/(97(-x+1)^{97)}-1/(49(-x+1)^{98)}+1/(99(-x+1)^{99)}+C

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integral of (x^2)/((x-1)^{100)}

\int \frac{x ^{2}}{( x - 1 ) ^{100}} d x

\frac{1}{97 ( - x + 1 ) ^{97}} - \frac{1}{49 ( - x + 1 ) ^{98}} + \frac{1}{99 ( - x + 1 ) ^{99}} + C

Solution steps

\int \frac{x ^{2}}{( x - 1 ) ^{100}} d x

Factor

x - 1 : - (- x + 1)

= \int \frac{x ^{2}}{( - ( - x + 1 ) ) ^{100}} d x

(- (- x + 1))^{100} = (- x + 1)^{100}

= \int \frac{x ^{2}}{( - x + 1 ) ^{100}} d x

Apply u-substitution

: \int - \frac{( - u + 1 ) ^{2}}{u ^{100}} d u

= \int - \frac{( - u + 1 ) ^{2}}{u ^{100}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{( - u + 1 ) ^{2}}{u ^{100}} d u

Expand

\frac{( - u + 1 ) ^{2}}{u ^{100}} : \frac{1}{u ^{98}} - \frac{2}{u ^{99}} + \frac{1}{u ^{100}}

= - \int \frac{1}{u ^{98}} - \frac{2}{u ^{99}} + \frac{1}{u ^{100}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int \frac{1}{u ^{98}} d u - \int \frac{2}{u ^{99}} d u + \int \frac{1}{u ^{100}} d u)

\int \frac{1}{u ^{98}} d u = - \frac{1}{97 u ^{97}}

\int \frac{2}{u ^{99}} d u = - \frac{1}{49 u ^{98}}

\int \frac{1}{u ^{100}} d u = - \frac{1}{99 u ^{99}}

= - (- \frac{1}{97 u ^{97}} - (- \frac{1}{49 u ^{98}}) - \frac{1}{99 u ^{99}})

Substitute back

u = - x + 1

= - (- \frac{1}{97 ( - x + 1 ) ^{97}} - (- \frac{1}{49 ( - x + 1 ) ^{98}}) - \frac{1}{99 ( - x + 1 ) ^{99}})

Simplify

= \frac{1}{97 ( - x + 1 ) ^{97}} - \frac{1}{49 ( - x + 1 ) ^{98}} + \frac{1}{99 ( - x + 1 ) ^{99}}

Add a constant to the solution

= \frac{1}{97 ( - x + 1 ) ^{97}} - \frac{1}{49 ( - x + 1 ) ^{98}} + \frac{1}{99 ( - x + 1 ) ^{99}} + C

\int \frac{2}{x ( ln ( x ) ) ^{3}} d x

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\int \frac{1}{( 1 + x ) ( 1 + x ^{2} )} d x

\int (\frac{5 x + 3}{( x + 2 ) ^{2} ( x - 1 ) ^{2}}) d x

\int (\frac{2 p + 3}{( p - 1 ) ^{2}}) d p

What is the integral of (x^2)/((x-1)^{100)} ?
The integral of (x^2)/((x-1)^{100)} is 1/(97(-x+1)^{97)}-1/(49(-x+1)^{98)}+1/(99(-x+1)^{99)}+C