What is the integral of (1+sqrt(x))/(-xsqrt(x)) ?

The integral of (1+sqrt(x))/(-xsqrt(x)) is 2/(sqrt(x))-ln|x|+C

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integral of (1+sqrt(x))/(-xsqrt(x))

\int \frac{1 + x}{- x x} d x

\frac{2}{x} - ln ∣ x ∣ + C

Solution steps

\int \frac{1 + x}{- x x} d x

Take the constant out:

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

= - \int \frac{1 + x}{x x} d x

Expand

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

= - \int \frac{1}{x x} + \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}{x x} d x + \int \frac{1}{x} d x)

\int \frac{1}{x x} d x = - \frac{2}{x}

\int \frac{1}{x} d x = ln ∣ x ∣

= - (- \frac{2}{x} + ln ∣ x ∣)

Simplify

= \frac{2}{x} - ln ∣ x ∣

Add a constant to the solution

= \frac{2}{x} - ln ∣ x ∣ + C

\int_{0}^{1} - (x^{2} - x) d x

x \to 3 lim (- 7 - x)

\int x e^{2 x} d x

y^{^{''''}} - 4 y^{^{''}} = 0

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

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