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

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

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

integral

\frac{1}{( 1 + 4 \frac{x}{l} )}

\frac{l}{4} ln 1 + \frac{4 x}{l} + C

Solution steps

\int \frac{1}{1 + 4 \cdot \frac{x}{l}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{l}{4} ln ∣ u ∣

Substitute back

u = 1 + 4 \cdot \frac{x}{l}

= \frac{l}{4} ln 1 + 4 \cdot \frac{x}{l}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{l}{4} ln 1 + \frac{4 x}{l}

Add a constant to the solution

= \frac{l}{4} ln 1 + \frac{4 x}{l} + C

\frac{2 x}{x ^{2}}

\frac{x ^{4} + x ^{2} + 1}{( x + 1 )}

\frac{2 x}{( x ^{2} + 1 )}

e^{x^{3}}

ln (\frac{( 9 + x )}{x})

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