What is the integral of 1/(sqrt(x)(4+x)) ?

The integral of 1/(sqrt(x)(4+x)) is arctan(1/2 sqrt(x))+C

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integral of 1/(sqrt(x)(4+x))

\int \frac{1}{x ( 4 + x )} d x

arctan (\frac{1}{2} x) + C

Solution steps

\int \frac{1}{x ( 4 + x )} d x

Apply u-substitution

: \int \frac{2}{4 + u ^{2}} d u

= \int \frac{2}{4 + u ^{2}} d u

Take the constant out:

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

= 2 \cdot \int \frac{1}{4 + u ^{2}} d u

Apply Integral Substitution

= 2 \cdot \int \frac{1}{2 ( v ^{2} + 1 )} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 2 \cdot \frac{1}{2} arctan (v)

Substitute back

= 2 \cdot \frac{1}{2} arctan (\frac{x}{2})

Simplify

2 \cdot \frac{1}{2} arctan (\frac{x}{2}) : arctan (\frac{1}{2} x)

= arctan (\frac{1}{2} x)

Add a constant to the solution

= arctan (\frac{1}{2} x) + C

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What is the integral of 1/(sqrt(x)(4+x)) ?
The integral of 1/(sqrt(x)(4+x)) is arctan(1/2 sqrt(x))+C