∫ e^(xlog 5) e^x dx , is: Where C is the constant of integration.

The correct answer is Option (1) → \(\displaystyle\frac{5^x}{\log 5}e^x+C\)

Note: The given answer is as per the official NTA Answer Key. However, the answer given by NTA appears to be incorrect, and the correct answer should be Option (4), as explained below.

Since \(e^{x\log 5} = e^{\log 5^x} = 5^x\), the integrand simplifies:

    \( \displaystyle\int e^{x\log 5}\, e^x\,dx = \int 5^x e^x\,dx = \int (5e)^x\,dx \)

Using \(\displaystyle\int a^x\,dx = \frac{a^x}{\log a} + C\) with \(a = 5e\):

    \( \displaystyle = \frac{(5e)^x}{\log(5e)} + C \)

Option (1) \(\displaystyle\frac{5^x}{\log 5}e^x+C\) — The answer given by NTA, but it seems to be incorrect. Differentiating it by the product rule:

    \( \displaystyle\frac{d}{dx}\left(\frac{5^x e^x}{\log 5}\right) = \frac{5^x \log 5 \cdot e^x + 5^x e^x}{\log 5} = 5^x e^x\left(1 + \frac{1}{\log 5}\right) \)

    This is not \(5^x e^x\).

Option (4) \(\displaystyle\frac{(5e)^x}{\log(5e)}+C\) — Should be correct.

Advice for the exam: In the real exam, mark the answer that is mathematically correct. If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.