What is the integral from 1 to e of (2x-3x^3)/(x^2) ?

The integral from 1 to e of (2x-3x^3)/(x^2) is 2-(3(e^2-1))/2

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integral from 1 to e of (2x-3x^3)/(x^2)

\int_{1}^{e} \frac{2 x - 3 x ^{3}}{x ^{2}} d x

2 - \frac{3 ( e ^{2} - 1 )}{2}

Decimal

- 7.58358 \dots

Solution steps

\int_{1}^{e} \frac{2 x - 3 x ^{3}}{x ^{2}} d x

Simplify

\frac{2 x - 3 x ^{3}}{x ^{2}} : \frac{2 - 3 x ^{2}}{x}

= \int_{1}^{e} \frac{2 - 3 x ^{2}}{x} d x

Expand

\frac{2 - 3 x ^{2}}{x} : \frac{2}{x} - 3 x

= \int_{1}^{e} \frac{2}{x} - 3 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{e} \frac{2}{x} d x - \int_{1}^{e} 3 x d x

\int_{1}^{e} \frac{2}{x} d x = 2

\int_{1}^{e} 3 x d x = \frac{3 ( e ^{2} - 1 )}{2}

= 2 - \frac{3 ( e ^{2} - 1 )}{2}

f (x) = x^{2} + 4 x

f (x) = 2 x - cot^{2} (x)

y = \frac{1}{x + 1}

f (x) = 1260 - 16 x^{2}, at x = 3

f (x) = x^{\frac{19}{9}} + x^{\frac{10}{9}}

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