What is the integral from 2 to 4 of (2x^2)/(x+2) ?

The integral from 2 to 4 of (2x^2)/(x+2) is 2(4ln(3/2)+2)

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integral from 2 to 4 of (2x^2)/(x+2)

\int_{2}^{4} \frac{2 x ^{2}}{x + 2} d x

2 (4 ln (\frac{3}{2}) + 2)

Decimal Notation

7.24372 \dots

Solution steps

\int_{2}^{4} \frac{2 x ^{2}}{x + 2} d x

Take the constant out:

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

= 2 \cdot \int_{2}^{4} \frac{x ^{2}}{x + 2} d x

Apply u-substitution

: \int_{4}^{6} \frac{( u - 2 ) ^{2}}{u} d u

= 2 \cdot \int_{4}^{6} \frac{( u - 2 ) ^{2}}{u} d u

Expand

\frac{( u - 2 ) ^{2}}{u} : u - 4 + \frac{4}{u}

= 2 \cdot \int_{4}^{6} u - 4 + \frac{4}{u} 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

= 2 (\int_{4}^{6} u d u - \int_{4}^{6} 4 d u + \int_{4}^{6} \frac{4}{u} d u)

\int_{4}^{6} u d u = 10

\int_{4}^{6} 4 d u = 8

\int_{4}^{6} \frac{4}{u} d u = 4 ln (\frac{3}{2})

= 2 (10 - 8 + 4 ln (\frac{3}{2}))

Simplify

= 2 (4 ln (\frac{3}{2}) + 2)

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What is the integral from 2 to 4 of (2x^2)/(x+2) ?
The integral from 2 to 4 of (2x^2)/(x+2) is 2(4ln(3/2)+2)