What is the integral of 1/(3x+7sqrt(x)) ?

The integral of 1/(3x+7sqrt(x)) is 2/3 ln|3sqrt(x)+7|+C

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integral of 1/(3x+7sqrt(x))

\int \frac{1}{3 x + 7 x} d x

\frac{2}{3} ln 3 x + 7 + C

Solution steps

\int \frac{1}{3 x + 7 x} d x

Apply u-substitution

: \int \frac{2}{3 u + 7} d u

= \int \frac{2}{3 u + 7} 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}{3 u + 7} d u

Apply u-substitution

: \int \frac{1}{3 v} d v

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 \cdot \frac{1}{3} ln ∣ v ∣

Substitute back

= 2 \cdot \frac{1}{3} ln 3 x + 7

Simplify

2 \cdot \frac{1}{3} ln 3 x + 7 : \frac{2}{3} ln 3 x + 7

= \frac{2}{3} ln 3 x + 7

Add a constant to the solution

= \frac{2}{3} ln 3 x + 7 + C

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What is the integral of 1/(3x+7sqrt(x)) ?
The integral of 1/(3x+7sqrt(x)) is 2/3 ln|3sqrt(x)+7|+C