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

The integral from 3 to 4 of (3x^2)/(x-2) is 3(4ln(2)+11/2)

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integral from 3 to 4 of (3x^2)/(x-2)

\int_{3}^{4} \frac{3 x ^{2}}{x - 2} d x

3 (4 ln (2) + \frac{11}{2})

Decimal

24.81776 \dots

Solution steps

\int_{3}^{4} \frac{3 x ^{2}}{x - 2} d x

Take the constant out:

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

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

Apply u-substitution

: \int_{1}^{2} \frac{( u + 2 ) ^{2}}{u} d u

= 3 \cdot \int_{1}^{2} \frac{( u + 2 ) ^{2}}{u} d u

Expand

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

= 3 \cdot \int_{1}^{2} 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

= 3 (\int_{1}^{2} u d u + \int_{1}^{2} 4 d u + \int_{1}^{2} \frac{4}{u} d u)

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

\int_{1}^{2} 4 d u = 4

\int_{1}^{2} \frac{4}{u} d u = 4 ln (2)

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

Combine the fractions

4 + \frac{3}{2} : \frac{11}{2}

= 3 (4 ln (2) + \frac{11}{2})

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What is the integral from 3 to 4 of (3x^2)/(x-2) ?
The integral from 3 to 4 of (3x^2)/(x-2) is 3(4ln(2)+11/2)