Question

\(\int \frac{30x^{14}+15^xlog225}{x^{15}+15^x}\,dx =\)

• A. \(x^{15}+15x^{14}+30x^{13}+c\), where c is the constant of integration
• B. \(\frac{log(x^{15}+15^x)}{2}+c\), where c is the constant of integration
• C. \(log(x^{15}+15^x)^2+c\), where c is the constant of integration
• D. \(log(x^{15}+15^x)+c\), where c is the constant of integration

Hint:

The numerator is a constant multiple of the derivative of the denominator.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
If the numerator is a multiple of the derivative of the denominator, the integral is that multiple of the log of the denominator: \(\int \dfrac{f'(x)}{f(x)}dx = \log|f(x)| + c\).

Step 2: Find the derivative of the denominator.
Let \(f(x) = x^{15} + 15^x\). Then \(f'(x) = 15x^{14} + 15^x \log 15\).

Step 3: Compare with the numerator.
\(\log 225 = \log 15^2 = 2\log 15\). So the numerator is \(30x^{14} + 2\cdot 15^x\log 15 = 2\left(15x^{14} + 15^x\log 15\right) = 2f'(x)\).

Step 4: Integrate.
\[ \int\frac{2f'(x)}{f(x)}dx = 2\log(x^{15} + 15^x) + c = \log(x^{15} + 15^x)^2 + c \]

Step 5: Check the options.
Option (D) misses the factor 2. Option (B) divides by 2 instead of multiplying. Option (A) is a polynomial and not a logarithm.

Final Answer:
The integral is \(\log(x^{15} + 15^x)^2 + c\), option (C). \[ \boxed{\log\left(x^{15}+15^x\right)^2 + c} \]