Question

Evaluate \( \int (\log x)^m x^n dx \).

• A. \( \int t^m e^{nt} dt, \quad t = e^x \)
• B. \( \int t^m e^{(n+1)t} dt, \quad t = e^x \)
• C. \( \int t^m e^{(n+1)t} dt, \quad x = e^t \)
• D. \( \int t^m e^{nt} dt, \quad x = e^t \)

Hint:

When evaluating integrals involving \(\log x\) and powers of \(x\), the substitution \(x = e^t\) is often useful.

Answer 1

The Correct Option is C

Let \( x = e^t \), so \( t = \log x \). Then,

\[ dx = e^t \, dt \] 

Substituting into the integral, we get:

\[ \int (\log x)^m x^n \, dx = \int t^m (e^t)^n e^t \, dt \] \[ = \int t^m e^{nt} e^t \, dt \] \[ = \int t^m e^{(n+1)t} \, dt \]

Therefore, the correct answer is:

\[ \int t^m e^{(n+1)t} \, dt, \quad x = e^t. \]