What is the integral of 5/(6x+5) ?

The integral of 5/(6x+5) is 5/6 ln|6x+5|+C

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integral of 5/(6x+5)

\int \frac{5}{6 x + 5} d x

\frac{5}{6} ln ∣ 6 x + 5 ∣ + C

Solution steps

\int \frac{5}{6 x + 5} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{6 x + 5} d x

Apply u-substitution

: \int \frac{1}{6 u} d u

= 5 \cdot \int \frac{1}{6 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{6} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{6} ln ∣ u ∣

Substitute back

u = 6 x + 5

= 5 \cdot \frac{1}{6} ln ∣ 6 x + 5 ∣

Simplify

5 \cdot \frac{1}{6} ln ∣ 6 x + 5 ∣ : \frac{5}{6} ln ∣ 6 x + 5 ∣

= \frac{5}{6} ln ∣ 6 x + 5 ∣

Add a constant to the solution

= \frac{5}{6} ln ∣ 6 x + 5 ∣ + C

\frac{x}{7 x ^{2} + 1}

f (x) = e^{2 e^{x}}

\frac{d}{d x} (e^{x} (a cos (x) + b sin (x)))

\int x^{6} e^{- x^{7}} d x

y = x^{2} - 4 x + 3, (1, 0)