Question

evaluate the integral \\(\int \frac{dx}{(6x + 10)^3}\\), by making the appropriate substitution: \\(u = \\) \\(\int \frac{dx}{(6x + 10)^3} = \\)

Explanation:

Step1: Choose substitution variable

Let $u = 6x + 10$

Step2: Find derivative of u

$\frac{du}{dx} = 6 \implies dx = \frac{1}{6}du$

Step3: Substitute into integral

$$\int \frac{\frac{1}{6}du}{u^3} = \frac{1}{6}\int u^{-3}du$$

Step4: Integrate using power rule

$\frac{1}{6} \cdot \frac{u^{-2}}{-2} + C = -\frac{1}{12}u^{-2} + C$

Step5: Substitute back u

$-\frac{1}{12(6x+10)^2} + C$

Answer:

Appropriate substitution: $u = 6x + 10$
Integral result: $\boldsymbol{-\frac{1}{12(6x+10)^2} + C}$ (where $C$ is the constant of integration)