What is the integral of 1/(x^2)sqrt(3+1/x) ?

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

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integral of 1/(x^2)sqrt(3+1/x)

\int \frac{1}{x ^{2}} 3 + \frac{1}{x} d x

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

Solution steps

\int \frac{1}{x ^{2}} 3 + \frac{1}{x} d x

Apply u-substitution

: \int - 3 + u d u

= \int - 3 + u d u

Take the constant out:

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

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

Join

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

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

Add a constant to the solution

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

\frac{d}{d x} (sin (ln (2 x)))

\int t sec^{2} (3 t) d t

\frac{d ^{2}}{d x ^{2}} (\frac{1}{2 + t})

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

\frac{d}{d x} (- cos^{3} (x))

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