What is the integral of 1/(4x+5sqrt(x)) ?

The integral of 1/(4x+5sqrt(x)) is 1/2 ln|4sqrt(x)+5|+C

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integral of 1/(4x+5sqrt(x))

\int \frac{1}{4 x + 5 x} d x

\frac{1}{2} ln 4 x + 5 + C

Solution steps

\int \frac{1}{4 x + 5 x} d x

Apply u-substitution

: \int \frac{2}{4 u + 5} d u

= \int \frac{2}{4 u + 5} 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 + 5} d u

Apply u-substitution

: \int \frac{1}{4 v} d v

= 2 \cdot \int \frac{1}{4 v} d v

Take the constant out:

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

= 2 \cdot \frac{1}{4} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 \cdot \frac{1}{4} ln ∣ v ∣

Substitute back

= 2 \cdot \frac{1}{4} ln 4 x + 5

Simplify

2 \cdot \frac{1}{4} ln 4 x + 5 : \frac{1}{2} ln 4 x + 5

= \frac{1}{2} ln 4 x + 5

Add a constant to the solution

= \frac{1}{2} ln 4 x + 5 + C

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\frac{d}{d x} (7 e^{- x} + e^{4 x})

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\frac{1}{2} x

What is the integral of 1/(4x+5sqrt(x)) ?
The integral of 1/(4x+5sqrt(x)) is 1/2 ln|4sqrt(x)+5|+C