The integral given is: ∫ex(1+tan‌x+tan2x)‌dx We can simplify the expression within the integral using a trigonometric identity. Recall the identity: sec2x=1+tan2x Using this identity, we can rewrite the integral as: ∫ex(sec2x+tan‌x)‌dx To solve this integral, we will use the method of substitution. Let's first focus on the part: ∫ex‌tan‌x‌dx We utilize integration by parts with the following choices: Let u=tan‌x, then du=sec2x‌dx. Let dv=ex‌dx, then v=ex. Now by integration by parts formula, ∫udv=uv−∫vdu, we get: ∫ex‌tan‌x‌dx=ex‌tan‌x−∫exsec2x‌dx We can reorganize the integral equation to: ‌∫ex(sec2x+tan‌x)‌dx=∫exsec2x‌dx+∫ex‌tan‌x‌dx ‌=∫exsec2x‌dx+ex‌tan‌x−∫exsec2x‌dx Here the term ∫exsec2x‌dx cancels out, leaving: ∫ex(sec2x+tan‌x)‌dx=ex‌tan‌x+C So, the integral evaluated is: ex‌tan‌x+C Option C is the correct answer: ex‌tan‌x+c