Question

\(\int \frac{(x+1)(x+logx)^2}{x}\,dx =\)

• A. \((\frac{x+logx}{x})^2+c\), where c is the constant of integration
• B. \(\frac{(x+logx)^2}{x}+c\), where c is the constant of integration
• C. \(\frac{(x+logx)^3}{3}+c\), where c is the constant of integration
• D. \(\frac{(x+logx)^3}{3x}+c\), where c is the constant of integration

Hint:

Notice that (x+1)/x dx is the differential of x + log x.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
If the integrand has the form \(g(u)\,u'\), substitute \(u\) as the inner function. Look at the derivative of \(x + \log x\).

Step 2: Spot the substitution:
\[ u = x + \log x \Rightarrow du = \left(1 + \frac1x\right)dx = \frac{x+1}{x}\,dx \]
The factor \(\dfrac{x+1}{x}dx\) in the integrand is exactly \(du\).

Step 3: Integrate:
\[ \int u^2\,du = \frac{u^3}{3} + c = \frac{(x + \log x)^3}{3} + c \]

Step 4: Check:
Differentiating the answer gives \((x + \log x)^2\left(1 + \tfrac1x\right) = \dfrac{(x+1)(x+\log x)^2}{x}\), the integrand. Options (A), (B) and (D) divide by \(x\), which would add a factor that is not in the derivative.

Final Answer:
The integral is (x + log x)^3 / 3 plus c. \[ \boxed{\text{(C) }\dfrac{(x+\log x)^3}{3}+c} \]