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

The integral from 0 to 2 of (2x)/((x^2+3)^3) is 20/441

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

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

\frac{20}{441}

Decimal

0.04535 \dots

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{3}^{7} \frac{1}{2 u ^{3}} d u

= 2 \cdot \int_{3}^{7} \frac{1}{2 u ^{3}} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int_{3}^{7} \frac{1}{u ^{3}} d u

Apply the Power Rule

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

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

Simplify

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

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

Compute the boundaries:

\frac{20}{441}

= \frac{20}{441}

x^{2} + x

sin (\frac{π x}{3})

\int \frac{( 4 + x + x ^{2} )}{x} d x

f (x) = ln (\frac{x ^{2} - 1}{x ^{2} + 1})

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

What is the integral from 0 to 2 of (2x)/((x^2+3)^3) ?
The integral from 0 to 2 of (2x)/((x^2+3)^3) is 20/441