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

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

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

\int_{1}^{e} \frac{( x ^{3} + 1 )}{x} d x

\frac{e ^{3} - 1}{3} + 1

Decimal

7.36184 \dots

Solution steps

\int_{1}^{e} \frac{x ^{3} + 1}{x} d x

Expand

\frac{x ^{3} + 1}{x} : x^{2} + \frac{1}{x}

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

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

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

= \frac{e ^{3} - 1}{3} + 1

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