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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\int_{0}^{1} x^{5} e^{- x^{6}} d x

\int_{0}^{8} 8 x + 36 d x

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