What is the integral of 1/(3sqrt(x)(2+\sqrt{x))} ?

The integral of 1/(3sqrt(x)(2+\sqrt{x))} is 2/3 ln|2+sqrt(x)|+C

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integral of 1/(3sqrt(x)(2+\sqrt{x))}

\int \frac{1}{3 x ( 2 + x )} d x

\frac{2}{3} ln 2 + x + C

Solution steps

\int \frac{1}{3 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}{3} \cdot \int \frac{1}{x ( 2 + x )} d x

Apply u-substitution

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

= \frac{1}{3} \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}{3} \cdot 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} \cdot 2 ln ∣ u ∣

Substitute back

u = 2 + x

= \frac{1}{3} \cdot 2 ln 2 + x

Simplify

\frac{1}{3} \cdot 2 ln 2 + x : \frac{2}{3} ln 2 + x

= \frac{2}{3} ln 2 + x

Add a constant to the solution

= \frac{2}{3} ln 2 + x + C

x \to 2 - lim (arcsin (\frac{x}{2}))

x \to - 1 lim ((x + 2 x^{4})^{3})

\int y^{x} d x

\int \frac{cos ^{4} ( x )}{sin ^{7} ( x )} d x

(1 + x) y^{^{'}} + y = cos (x)

What is the integral of 1/(3sqrt(x)(2+\sqrt{x))} ?
The integral of 1/(3sqrt(x)(2+\sqrt{x))} is 2/3 ln|2+sqrt(x)|+C