What is the integral from 1 to 2 of x/((x+2)^3) ?

The integral from 1 to 2 of x/((x+2)^3) is 5/144

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integral from 1 to 2 of x/((x+2)^3)

\int_{1}^{2} \frac{x}{( x + 2 ) ^{3}} d x

\frac{5}{144}

Decimal Notation

0.03472 \dots

Solution steps

\int_{1}^{2} \frac{x}{( x + 2 ) ^{3}} d x

Apply u-substitution

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

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

Expand

\frac{u - 2}{u ^{3}} : \frac{1}{u ^{2}} - \frac{2}{u ^{3}}

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

\int_{3}^{4} \frac{1}{u ^{2}} d u = \frac{1}{12}

\int_{3}^{4} \frac{2}{u ^{3}} d u = \frac{7}{144}

= \frac{1}{12} - \frac{7}{144}

Simplify

\frac{1}{12} - \frac{7}{144} : \frac{5}{144}

= \frac{5}{144}

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What is the integral from 1 to 2 of x/((x+2)^3) ?
The integral from 1 to 2 of x/((x+2)^3) is 5/144