Question

$\int\frac{\log(1+x)}{(1+x)}dx=$ ________.

• A. $\frac{1}{2}\log(1+x)+C$
• B. $\frac{1}{2}[\log(1+x)]^{2}+C$
• C. $[\log(1+x)]^{2}+C$
• D. $\log(1+x)+C$
• E. $x \log(1+x)+C$

Hint:

If an integral is of the form $\int f(x) f'(x) dx$, the answer is $\frac{[f(x)]^2}{2} + C$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the method of substitution.

Step 2: Meaning
Let $u = \log(1+x)$, then $du = \frac{1}{1+x}dx$.

Step 3: Analysis
The integral becomes $\int u du$.

Step 4: Conclusion
$\int u du = \frac{u^2}{2} + C = \frac{1}{2}[\log(1+x)]^2 + C$. Final Answer: (B)