1. **State the problem:** We need to solve the integral $$\int x^4 (3 - 5x^5)^{\frac{1}{3}} \, dx$$ using the method of substitution. 2. **Identify substitution:** Let $$u = 3 - 5x^5$$. This substitution is chosen because the expression inside the cube root is complicated and involves $x^5$. 3. **Compute differential:** Differentiate $u$ with respect to $x$: $$\frac{du}{dx} = -25x^4$$ So, $$du = -25x^4 \, dx$$ which implies $$x^4 \, dx = -\frac{1}{25} du$$ 4. **Rewrite the integral:** Substitute $u$ and $x^4 dx$ into the integral: $$\int x^4 (3 - 5x^5)^{\frac{1}{3}} \, dx = \int (u)^{\frac{1}{3}} \left(-\frac{1}{25} du\right) = -\frac{1}{25} \int u^{\frac{1}{3}} \, du$$ 5. **Integrate:** Use the power rule for integration: $$\int u^{\frac{1}{3}} \, du = \frac{u^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} = \frac{u^{\frac{4}{3}}}{\frac{4}{3}} = \frac{3}{4} u^{\frac{4}{3}}$$ 6. **Combine constants:** $$-\frac{1}{25} \times \frac{3}{4} u^{\frac{4}{3}} = -\frac{3}{100} u^{\frac{4}{3}} + C$$ 7. **Back-substitute:** Replace $u$ with original expression: $$-\frac{3}{100} (3 - 5x^5)^{\frac{4}{3}} + C$$ **Final answer:** $$\boxed{-\frac{3}{100} (3 - 5x^5)^{\frac{4}{3}} + C}$$