Value of ∫≤ft(e^2log x+e^xlog2+e^2log2)dx is

The correct answer is Option (2) → \( \displaystyle\frac{x^{3}}{3}+\displaystyle\frac{2^{x}}{\log2}+4x+C \): where \( C \) is an arbitrary constant

Since \( e^{\log a}=a \), each term simplifies:

    \( e^{2\log x}=e^{\log x^{2}}=x^{2} \)
    \( e^{x\log2}=e^{\log 2^{x}}=2^{x} \)
    \( e^{2\log2}=e^{\log4}=4 \)

So

    \( \displaystyle \int\left(x^{2}+2^{x}+4\right)dx \)

    \( \displaystyle =\frac{x^{3}}{3}+\frac{2^{x}}{\log2}+4x+C \)

Why the other options are incorrect

• Option (1) \( \displaystyle\frac{x^{3}}{3}+2^{x}\log2+4x+C \): where \( C \) is an arbitrary constant — Incorrect. \( 2^{x}\log2 \) is the derivative of \( 2^{x} \), not its integral.

• Option (3) \( \displaystyle\frac{x^{3}}{3}+2^{x}\log2+C \): where \( C \) is an arbitrary constant — Incorrect. \( 2^{x}\log2 \) is the derivative of \( 2^{x} \), not its integral, and the term 4x is missing.

• Option (4) \( \displaystyle\frac{x^{3}}{3}+\displaystyle\frac{2^{x}}{\log2}+C \): where \( C \) is an arbitrary constant — Incorrect. The term 4x, the integral of \( e^{2\log2}=4 \), is missing.