(4) \( \int \frac{d x}{(3 x+5)^{4}} \)

To evaluate the integral: \[ \int \frac{dx}{(3x + 5)^4} \] **Step 1: Substitution** Let’s make a substitution to simplify the integral. Let: \[ u = 3x + 5 \] Then, compute the differential \( du \): \[ \frac{du}{dx} = 3 \quad \Rightarrow \quad dx = \frac{du}{3} \] **Step 2: Rewrite the Integral in Terms of \( u \)** Substitute \( u \) and \( dx \) into the original integral: \[ \int \frac{dx}{(3x + 5)^4} = \int \frac{\frac{du}{3}}{u^4} = \frac{1}{3} \int u^{-4} \, du \] **Step 3: Integrate** Integrate \( u^{-4} \) with respect to \( u \): \[ \frac{1}{3} \int u^{-4} \, du = \frac{1}{3} \left( \frac{u^{-3}}{-3} \right) + C = -\frac{1}{9} u^{-3} + C \] **Step 4: Substitute Back to \( x \)** Replace \( u \) with the original expression \( 3x + 5 \): \[ -\frac{1}{9} u^{-3} + C = -\frac{1}{9(3x + 5)^3} + C \] **Final Answer:** \[ \int \frac{dx}{(3x + 5)^4} = -\frac{1}{9(3x + 5)^3} + C \] where \( C \) is the constant of integration.