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

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

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

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

\frac{1}{4} ln ∣ 4 x + 5 ∣ + C

Solution steps

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

Apply u-substitution

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

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = 4 x + 5

= \frac{1}{4} ln ∣ 4 x + 5 ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ 4 x + 5 ∣ + C

\int sin (2 x) cos^{2} (2 x) d x

x \frac{d y}{d x} = y - x^{2} + y^{2}

\frac{d}{d x} (\frac{10}{2 x})

\frac{d}{d x} (\frac{x}{2} ln (\frac{1 + x}{1 - x}) - 1)

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