What is the integral of 1/(5sqrt(x)(2+\sqrt{x))} ?

The integral of 1/(5sqrt(x)(2+\sqrt{x))} is 2/5 ln|2+sqrt(x)|+C

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integral of 1/(5sqrt(x)(2+\sqrt{x))}

\int \frac{1}{5 x ( 2 + x )} d x

\frac{2}{5} ln 2 + x + C

Solution steps

\int \frac{1}{5 x ( 2 + x )} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{x ( 2 + x )} d x

Apply u-substitution

: \int \frac{2}{u} d u

= \frac{1}{5} \cdot \int \frac{2}{u} d u

Take the constant out:

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

= \frac{1}{5} \cdot 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} \cdot 2 ln ∣ u ∣

Substitute back

u = 2 + x

= \frac{1}{5} \cdot 2 ln 2 + x

Simplify

\frac{1}{5} \cdot 2 ln 2 + x : \frac{2}{5} ln 2 + x

= \frac{2}{5} ln 2 + x

Add a constant to the solution

= \frac{2}{5} ln 2 + x + C

\frac{\partial}{\partial x} (e^{x + y + 4})

x \to 5 lim (x - 5)

f (x) = x^{3} - 2 x^{2}

x \to 1 lim (5 x + 4)

k^{2} + 7 x + 10

What is the integral of 1/(5sqrt(x)(2+\sqrt{x))} ?
The integral of 1/(5sqrt(x)(2+\sqrt{x))} is 2/5 ln|2+sqrt(x)|+C