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

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

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

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

\frac{3}{7} ln ∣ 1 + 7 x ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{7 u} d u

= 3 \cdot \int \frac{1}{7 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{7} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{7} ln ∣ u ∣

Substitute back

u = 1 + 7 x

= 3 \cdot \frac{1}{7} ln ∣ 1 + 7 x ∣

Simplify

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

= \frac{3}{7} ln ∣ 1 + 7 x ∣

Add a constant to the solution

= \frac{3}{7} ln ∣ 1 + 7 x ∣ + C

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