1. **State the problem:** Evaluate the integral $$\int (2x^2)(\sec^2 x)(\tan x) \, dx$$. 2. **Recall relevant formulas and rules:** - The derivative of $$\tan x$$ is $$\sec^2 x$$. - Integration by parts or substitution might be useful. 3. **Rewrite the integral:** $$\int 2x^2 \sec^2 x \tan x \, dx$$. 4. **Consider substitution:** Let $$u = \tan x$$, then $$du = \sec^2 x \, dx$$. 5. **Rewrite integral in terms of $$u$$:** $$\int 2x^2 u \, du$$. 6. **Problem:** The variable $$x$$ is still present, so substitution is not straightforward. 7. **Alternative approach:** Treat $$x$$ as a variable independent of $$u$$, so integration by parts is needed. 8. **Integration by parts formula:** $$\int f(x) g'(x) dx = f(x) g(x) - \int f'(x) g(x) dx$$. 9. **Set:** - $$f(x) = 2x^2$$ - $$g'(x) = \sec^2 x \tan x$$ 10. **Find $$g(x)$$:** Note that $$\frac{d}{dx} (\sec x) = \sec x \tan x$$. 11. **Rewrite $$g'(x)$$:** $$\sec^2 x \tan x = \sec x (\sec x \tan x) = \sec x \frac{d}{dx} (\sec x)$$. 12. **Integrate $$g'(x)$$:** $$g(x) = \int \sec x \frac{d}{dx} (\sec x) dx = \frac{1}{2} (\sec x)^2 + C$$ (using substitution). 13. **Apply integration by parts:** $$\int 2x^2 \sec^2 x \tan x dx = 2x^2 \cdot \frac{1}{2} \sec^2 x - \int \frac{d}{dx} (2x^2) \cdot \frac{1}{2} \sec^2 x dx$$ 14. **Simplify:** $$= x^2 \sec^2 x - \int 4x \cdot \frac{1}{2} \sec^2 x dx = x^2 \sec^2 x - 2 \int x \sec^2 x dx$$ 15. **Now evaluate $$\int x \sec^2 x dx$$ using integration by parts again:** - Let $$u = x$$, $$dv = \sec^2 x dx$$ - Then $$du = dx$$, $$v = \tan x$$ 16. **Apply integration by parts:** $$\int x \sec^2 x dx = x \tan x - \int \tan x dx$$ 17. **Integrate $$\int \tan x dx$$:** $$\int \tan x dx = -\ln |\cos x| + C$$ 18. **So:** $$\int x \sec^2 x dx = x \tan x + \ln |\cos x| + C$$ 19. **Substitute back:** $$\int 2x^2 \sec^2 x \tan x dx = x^2 \sec^2 x - 2 (x \tan x + \ln |\cos x|) + C$$ 20. **Final answer:** $$\boxed{x^2 \sec^2 x - 2x \tan x - 2 \ln |\cos x| + C}$$