What is the integral of 1/(x^2)sqrt(2+1/x) ?

The integral of 1/(x^2)sqrt(2+1/x) is -2/3 ((2x+1)/x)^{3/2}+C

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integral of 1/(x^2)sqrt(2+1/x)

\int \frac{1}{x ^{2}} 2 + \frac{1}{x} d x

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

Solution steps

\int \frac{1}{x ^{2}} 2 + \frac{1}{x} d x

Apply u-substitution

: \int - 2 + u d u

= \int - 2 + u d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int 2 + u d u

Apply u-substitution

: \int v d v

= - \int v d v

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

= - \frac{2}{3} v^{\frac{3}{2}}

Substitute back

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

Join

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

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

Add a constant to the solution

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

\int \frac{( x ^{- 2} - 4 )}{x ^{3}} d x

\int (2 x - 3)^{2} (2 x - 3)^{2} + 4 d x

cos^{2} (17 t)

x \to \infty lim (\frac{x ^{l n (n)}}{x ^{l n (n) + 1}})

ln (\frac{x}{y})

What is the integral of 1/(x^2)sqrt(2+1/x) ?
The integral of 1/(x^2)sqrt(2+1/x) is -2/3 ((2x+1)/x)^{3/2}+C