∫_1^4 x^3log x dx is equal to:

The correct answer is Option (3) → \( 128\log 2-\displaystyle\frac{255}{16} \)

Here log x is the natural logarithm. By integration by parts, with log x as the first function (it becomes simpler on differentiation):

    \( \displaystyle \int x^{3}\log x\,dx=\log x\cdot\frac{x^{4}}{4}-\int\frac{1}{x}\cdot\frac{x^{4}}{4}\,dx \)

    \( \displaystyle =\frac{x^{4}}{4}\log x-\frac{x^{4}}{16} \)

Applying the limits:

    \( \displaystyle \int_{1}^{4}x^{3}\log x\,dx=\left(\frac{256}{4}\log 4-\frac{256}{16}\right)-\left(\frac{1}{4}\log 1-\frac{1}{16}\right) \)

    \( \displaystyle =64\log 4-\frac{256}{16}+\frac{1}{16} \)

    \( \displaystyle =64\log 4-\frac{255}{16} \)

Since \( 64\log 4=64\cdot 2\log 2=128\log 2 \), the value is \( 128\log 2-\frac{255}{16} \).

Why the other options are incorrect

• Option (1) \( 64\log 4-\displaystyle\frac{256}{16} \) — Incorrect. The lower limit has been left out: at x = 1 the term \( -\frac{x^{4}}{16} \) is \( -\frac{1}{16} \), which turns \( -\frac{256}{16} \) into \( -\frac{255}{16} \).

• Option (2) \( 128\log 2+\displaystyle\frac{255}{16} \) — Incorrect. The sign of \( \frac{255}{16} \) is wrong; the \( \frac{x^{4}}{16} \) term is subtracted.

• Option (4) \( 64\log 2-\displaystyle\frac{255}{4} \) — Incorrect. \( 64\log 4 \) equals \( 128\log 2 \), not \( 64\log 2 \), and the constant is \( \frac{255}{16} \), not \( \frac{255}{4} \).