What is the integral of 1/(6sqrt(x)(5+\sqrt{x))} ?

The integral of 1/(6sqrt(x)(5+\sqrt{x))} is 1/3 ln|5+sqrt(x)|+C

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integral of 1/(6sqrt(x)(5+\sqrt{x))}

\int \frac{1}{6 x ( 5 + x )} d x

\frac{1}{3} ln 5 + x + C

Solution steps

\int \frac{1}{6 x ( 5 + x )} d x

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{x ( 5 + x )} d x

Apply u-substitution

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

= \frac{1}{6} \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}{6} \cdot 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{6} \cdot 2 ln ∣ u ∣

Substitute back

u = 5 + x

= \frac{1}{6} \cdot 2 ln 5 + x

Simplify

\frac{1}{6} \cdot 2 ln 5 + x : \frac{1}{3} ln 5 + x

= \frac{1}{3} ln 5 + x

Add a constant to the solution

= \frac{1}{3} ln 5 + x + C

6 y^{^{''}} + 31 y = 0

x \to 4 lim (\frac{x ^{3} - 6}{x - 4})

x \to - 3 lim (\frac{x}{x - 3})

f (x) = - 6 x^{3}

x \to 1 lim (\frac{6 x ^{2} + 12 x - 18}{2 x - 2})

What is the integral of 1/(6sqrt(x)(5+\sqrt{x))} ?
The integral of 1/(6sqrt(x)(5+\sqrt{x))} is 1/3 ln|5+sqrt(x)|+C