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

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

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integral of 1/(x^2)(1+1/x)^{1/2}

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

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

Solution steps

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

Apply u-substitution

: \int - u^{\frac{1}{2}} d u

= \int - u^{\frac{1}{2}} d u

Take the constant out:

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

= - \int u^{\frac{1}{2}} d u

Apply the Power Rule

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

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

Substitute back

u = 1 + \frac{1}{x}

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

Join

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

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

Add a constant to the solution

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

2 y (x + 1) d y = x d x

\frac{1}{x ^{2}}, at 4

\frac{\partial}{\partial x} (e^{x} x + 2 e^{x} y)

\int_{2}^{3} 3 - x d x

\frac{d}{d x} (ln (x) csc (x))

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