1. **State the problem:** We need to find the indefinite integral $$\int x^5 e^{x^6 + 1} \, dx$$. 2. **Identify substitution:** Let $$u = x^6 + 1$$. Then, differentiate: $$\frac{du}{dx} = 6x^5 \implies du = 6x^5 \, dx$$. 3. **Rewrite the integral:** From the substitution, we have: $$x^5 \, dx = \frac{du}{6}$$. So the integral becomes: $$\int x^5 e^{x^6 + 1} \, dx = \int e^u \frac{du}{6} = \frac{1}{6} \int e^u \, du$$. 4. **Integrate:** The integral of $$e^u$$ with respect to $$u$$ is $$e^u$$, so: $$\frac{1}{6} \int e^u \, du = \frac{1}{6} e^u + C$$. 5. **Back-substitute:** Replace $$u$$ with $$x^6 + 1$$: $$\frac{1}{6} e^{x^6 + 1} + C$$. 6. **Final answer:** The integral is $$\int x^5 e^{x^6 + 1} \, dx = \frac{1}{6} e^{x^6 + 1} + C$$. **Check options:** - (a) $$\frac{x^5}{6} e^{x^6 + 1}$$ is incorrect. - (b) $$-\frac{1}{6} e^{x^6 + 1}$$ is incorrect (wrong sign). - (c) $$\frac{1}{6} x^6 e^{6x^5}$$ is incorrect. - (d) $$\frac{x^5}{6} e^{\frac{7}{x} + x}$$ is incorrect. **Correct answer is none of the given options exactly, but the integral evaluates to** $$\frac{1}{6} e^{x^6 + 1} + C$$.