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

The integral of ((x+1)/x)^3 is ((x+1)^3)/(x^2)+3(ln|x|-2/x-1/(2x^2))+C

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integral of ((x+1)/x)^3

integral

(\frac{x + 1}{x})^{3}

\frac{( x + 1 ) ^{3}}{x ^{2}} + 3 (ln ∣ x ∣ - \frac{2}{x} - \frac{1}{2 x ^{2}}) + C

Solution steps

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \int \frac{( x + 1 ) ^{3}}{x ^{3}} d x

Apply Integration By Parts

: \frac{( x + 1 ) ^{3}}{x ^{2}} - \int - \frac{3 ( x + 1 ) ^{2}}{x ^{3}} d x

= \frac{( x + 1 ) ^{3}}{x ^{2}} - \int - \frac{3 ( x + 1 ) ^{2}}{x ^{3}} d x

\int - \frac{3 ( x + 1 ) ^{2}}{x ^{3}} d x = - 3 (ln ∣ x ∣ - \frac{2}{x} - \frac{1}{2 x ^{2}})

= \frac{( x + 1 ) ^{3}}{x ^{2}} - (- 3 (ln ∣ x ∣ - \frac{2}{x} - \frac{1}{2 x ^{2}}))

Simplify

= \frac{( x + 1 ) ^{3}}{x ^{2}} + 3 (ln ∣ x ∣ - \frac{2}{x} - \frac{1}{2 x ^{2}})

Add a constant to the solution

= \frac{( x + 1 ) ^{3}}{x ^{2}} + 3 (ln ∣ x ∣ - \frac{2}{x} - \frac{1}{2 x ^{2}}) + C

\frac{- 1}{x ^{2}}

e^{- 2} x

e^{3} x

\frac{e ^{x}}{2}

1 + x

What is the integral of ((x+1)/x)^3 ?
The integral of ((x+1)/x)^3 is ((x+1)^3)/(x^2)+3(ln|x|-2/x-1/(2x^2))+C