Question

\(\displaystyle \int (\log x)^2 x^3\,dx=\dfrac{x^4}{32}f(x)+C \Rightarrow f(x)=\)

• A. \(8(\log x)^2-4\log x+1\)
• B. \(8\log x-4x^4+x^3\)
• C. \(8(\log x)^2+4x-x^2\)
• D. \(4(\log x)^2-4x^2+x+1\)

Hint:

For integrals of the form \[ \int x^n(\log x)^m\,dx, \] use integration by parts repeatedly by taking the logarithmic expression as the first function.

Answer 1

The Correct Option is A

Step 1: Apply the tabular integration by parts method.
Let u = (log x)², dv = x³ dx. Differentiate u repeatedly: u' = 2(log x)/x, u'' = 2(1–log x)/x². Integrate dv repeatedly: ∫x³dx = x⁴/4, ∫x⁴/4 dx = x⁵/20.

Step 2: Build the tabular table and alternate signs.
Row 1: (log x)² · x⁴/4. Row 2: –[2(log x)/x] · x⁵/20 = –(log x)x⁴/10. Row 3: +[2(1–log x)/x²] · x⁶/120 = (1–log x)x⁴/60.

Step 3: Combine and simplify.
Integral = x⁴/4 (log x)² – x⁴/10 log x + x⁴/60 (1–log x) + C = x⁴[ (1/4)(log x)² – (1/10)log x + 1/60 – (1/60)log x ] + C = x⁴[ (1/4)(log x)² – (1/8)log x + 1/32 ]·(32/32) + C = (x⁴/32)[8(log x)² – 4 log x + 1] + C.

Step 4: Final Answer:
f(x) = 8(log x)² – 4 log x + 1.