Question

consider the integral below.$int \frac{7}{9x + 3}dx$which integral corresponds to a correct $u$-substitution?$\bigcirc int \frac{7}{9x}du$$\bigcirc int \frac{7}{9x}dx$$\bigcirc int \frac{7}{9u + 3}du$$\bigcirc int \frac{7}{9u}du$$\bigcirc int \frac{7}{9u}dx$$\bigcirc int \frac{63}{u}du$$\bigcirc int \frac{7}{u}du$$\bigcirc$ none of the above

Explanation:

Step1: Choose substitution variable

Let $u = 9x + 3$

Step2: Find derivative of u

$\frac{du}{dx} = 9 \implies dx = \frac{1}{9}du$

Step3: Substitute into original integral

Substitute $u = 9x+3$ and $dx=\frac{1}{9}du$:
$$\int \frac{7}{u} \cdot \frac{1}{9} du = \int \frac{7}{9u} du$$

Answer:

$\boldsymbol{\int \frac{7}{9u} du}$