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

The integral from 0 to 2 of (2x)/((x^2+2)^2) is 1/3

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

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

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

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

Apply u-substitution

= 2 \cdot \int_{2}^{6} \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

= 2 \cdot \frac{1}{2} \cdot \int_{2}^{6} \frac{1}{u ^{2}} d u

Apply the Power Rule

= 2 \cdot \frac{1}{2} [- \frac{1}{u}]_{2}^{6}

Simplify

2 \cdot \frac{1}{2} [- \frac{1}{u}]_{2}^{6} : [- \frac{1}{u}]_{2}^{6}

= [- \frac{1}{u}]_{2}^{6}

Compute the boundaries:

\frac{1}{3}

= \frac{1}{3}

What is the integral from 0 to 2 of (2x)/((x^2+2)^2) ?
The integral from 0 to 2 of (2x)/((x^2+2)^2) is 1/3