What is the integral from-2 to 0 of x/((x^2+3)^2) ?

The integral from-2 to 0 of x/((x^2+3)^2) is -2/21

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integral from-2 to 0 of x/((x^2+3)^2)

\int_{- 2}^{0} \frac{x}{( x ^{2} + 3 ) ^{2}} d x

- \frac{2}{21}

Decimal

- 0.09523 \dots

Solution steps

\int_{- 2}^{0} \frac{x}{( x ^{2} + 3 ) ^{2}} d x

Apply u-substitution

= \int_{7}^{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}^{7} \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}^{7} \frac{1}{u ^{2}} d u

Apply the Power Rule

= - \frac{1}{2} [- \frac{1}{u}]_{3}^{7}

Compute the boundaries:

\frac{4}{21}

= - \frac{1}{2} \cdot \frac{4}{21}

Simplify

= - \frac{2}{21}

(\frac{1}{4 x})^{^{'}}

\frac{d}{d x} (4 x + 8 x^{\frac{3}{4}})

x \to 2 lim ([f (x) + 5 g (x)])

(2^{x})^{^{'}}

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

What is the integral from-2 to 0 of x/((x^2+3)^2) ?
The integral from-2 to 0 of x/((x^2+3)^2) is -2/21