Question

If ∫(log x)³ x⁵ dx = \(\frac{x^6}{A}\) [B(log x)³ + C(logx)² + D(log x) - 1] + k and A,B,C,D are integers, then A - (B+C+D) = 

• A. 172
• B. 184
• C. 192
• D. 216

Answer 1

The Correct Option is C

To solve the problem, we need to evaluate the integral $I = \int (\log x)^3 x^5 dx$ and find $A - (B + C + D)$ for the form $\frac{x^6}{A} [B (\log x)^3 + C (\log x)^2 + D \log x - 1] + k$.

1. First Integration by Parts:
Let $u = (\log x)^3$, $dv = x^5 dx$.
Then $du = 3 (\log x)^2 \cdot \frac{1}{x} dx$, $v = \frac{x^6}{6}$.
$I = \frac{x^6}{6} (\log x)^3 - \int \frac{x^6}{6} \cdot 3 (\log x)^2 \cdot \frac{1}{x} dx = \frac{x^6}{6} (\log x)^3 - \frac{1}{2} \int x^5 (\log x)^2 dx$.

2. Second Integration by Parts:
For $\int x^5 (\log x)^2 dx$, let $u = (\log x)^2$, $dv = x^5 dx$.
Then $du = 2 \log x \cdot \frac{1}{x} dx$, $v = \frac{x^6}{6}$.
$\int x^5 (\log x)^2 dx = \frac{x^6}{6} (\log x)^2 - \int \frac{x^6}{6} \cdot 2 \log x \cdot \frac{1}{x} dx = \frac{x^6}{6} (\log x)^2 - \frac{1}{3} \int x^5 \log x dx$.

3. Third Integration by Parts:
For $\int x^5 \log x dx$, let $u = \log x$, $dv = x^5 dx$.
Then $du = \frac{1}{x} dx$, $v = \frac{x^6}{6}$.
$\int x^5 \log x dx = \frac{x^6}{6} \log x - \int \frac{x^6}{6} \cdot \frac{1}{x} dx = \frac{x^6}{6} \log x - \frac{1}{6} \cdot \frac{x^6}{6} = \frac{x^6}{6} \log x - \frac{x^6}{36}$.

4. Combine Results:
Substitute back:
$\int x^5 (\log x)^2 dx = \frac{x^6}{6} (\log x)^2 - \frac{1}{3} \left( \frac{x^6}{6} \log x - \frac{x^6}{36} \right) = \frac{x^6}{6} (\log x)^2 - \frac{x^6}{18} \log x + \frac{x^6}{108}$.
Then:
$I = \frac{x^6}{6} (\log x)^3 - \frac{1}{2} \left( \frac{x^6}{6} (\log x)^2 - \frac{x^6}{18} \log x + \frac{x^6}{108} \right)$.
Simplify:
$I = \frac{x^6}{6} (\log x)^3 - \frac{x^6}{12} (\log x)^2 + \frac{x^6}{36} \log x - \frac{x^6}{216} = \frac{x^6}{216} [36 (\log x)^3 - 18 (\log x)^2 + 6 \log x - 1]$.

5. Identify Coefficients:
Compare with $\frac{x^6}{A} [B (\log x)^3 + C (\log x)^2 + D \log x - 1] + k$:
$A = 216$, $B = 36$, $C = -18$, $D = 6$.

6. Compute the Final Expression:
Calculate $A - (B + C + D)$:
$B + C + D = 36 - 18 + 6 = 24$.
$A - (B + C + D) = 216 - 24 = 192$.

Final Answer:
The value of $A - (B + C + D)$ is $192$.