What is the integral from 0 to x of 1/((1-x)(1-x)) ?

The integral from 0 to x of 1/((1-x)(1-x)) is 1/(1-x)-1

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integral from 0 to x of 1/((1-x)(1-x))

\int_{0}^{x} \frac{1}{( 1 - x ) ( 1 - x )} d x

\frac{1}{1 - x} - 1

Solution steps

\int_{0}^{x} \frac{1}{( 1 - x ) ( 1 - x )} d x

(1 - x) (1 - x) = (1 - x)^{2}

= \int_{0}^{x} \frac{1}{( 1 - x ) ^{2}} d x

Apply u-substitution

: \int_{1}^{1 - x} - \frac{1}{u ^{2}} d u

= \int_{1}^{1 - x} - \frac{1}{u ^{2}} d u

Take the constant out:

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

= - \int_{1}^{1 - x} \frac{1}{u ^{2}} d u

Apply the Power Rule

: [- \frac{1}{u}]_{1}^{1 - x}

= - [- \frac{1}{u}]_{1}^{1 - x}

Compute the boundaries:

- \frac{1}{1 - x} + 1

= - (- \frac{1}{1 - x} + 1)

Simplify

= \frac{1}{1 - x} - 1

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What is the integral from 0 to x of 1/((1-x)(1-x)) ?
The integral from 0 to x of 1/((1-x)(1-x)) is 1/(1-x)-1