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

The integral from 1/e to e^2 of (x^3+1)/x is (e^6)/3-1/(3e^3)+3

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

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

\frac{e ^{6}}{3} - \frac{1}{3 e ^{3}} + 3

Decimal

137.45966 \dots

Solution steps

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

Expand

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

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

\int_{\frac{1}{e}}^{e^{2}} x^{2} d x = \frac{e ^{6}}{3} - \frac{1}{3 e ^{3}}

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

= \frac{e ^{6}}{3} - \frac{1}{3 e ^{3}} + 3

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What is the integral from 1/e to e^2 of (x^3+1)/x ?
The integral from 1/e to e^2 of (x^3+1)/x is (e^6)/3-1/(3e^3)+3