What is the integral from 0 to 1 of x/((4-x^2)^2) ?

The integral from 0 to 1 of x/((4-x^2)^2) is 1/24

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integral from 0 to 1 of x/((4-x^2)^2)

\int_{0}^{1} \frac{x}{( 4 - x ^{2} ) ^{2}} d x

\frac{1}{24}

Decimal Notation

0.04166 \dots

Solution steps

\int_{0}^{1} \frac{x}{( 4 - x ^{2} ) ^{2}} d x

Apply u-substitution

: \int_{4}^{3} - \frac{1}{2 u ^{2}} d u

= \int_{4}^{3} - \frac{1}{2 u ^{2}} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{3}^{4} - \frac{1}{2 u ^{2}} d u

Take the constant out:

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

= - (- \frac{1}{2} \cdot \int_{3}^{4} \frac{1}{u ^{2}} d u)

Apply the Power Rule

: [- \frac{1}{u}]_{3}^{4}

= - (- \frac{1}{2} [- \frac{1}{u}]_{3}^{4})

Simplify

= \frac{1}{2} [- \frac{1}{u}]_{3}^{4}

Compute the boundaries:

\frac{1}{12}

= \frac{1}{2} \cdot \frac{1}{12}

Simplify

= \frac{1}{24}

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What is the integral from 0 to 1 of x/((4-x^2)^2) ?
The integral from 0 to 1 of x/((4-x^2)^2) is 1/24