What is the integral of 1/(5+3x) ?

The integral of 1/(5+3x) is 1/3 ln|5+3x|+C

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integral of 1/(5+3x)

\int \frac{1}{5 + 3 x} d x

\frac{1}{3} ln ∣ 5 + 3 x ∣ + C

Solution steps

\int \frac{1}{5 + 3 x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{3} ln ∣ u ∣

Substitute back

u = 5 + 3 x

= \frac{1}{3} ln ∣ 5 + 3 x ∣

Add a constant to the solution

= \frac{1}{3} ln ∣ 5 + 3 x ∣ + C

\frac{d}{d x} ((1 + \frac{3}{x})^{x})

\frac{\partial}{\partial x} (\frac{( 2 x - 5 y )}{2 x + 5 y})

\int_{0}^{π} 6 sin^{4} (x) d x

y = r^{2} - x^{2}, y = 0