Maths · Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions
d x= \)
\( \int\left(\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}}\right) d x= \)
- A. \( x+c \)
- B. \( 3 x^{2}+c \)
- C. \( \frac{x^{3}}{3}+c \)
- D. \( \frac{x^{3}}{2}+c \)
Step-by-step solution
Simplify using \( e^{k \log x} = x^k \). Numerator: \( x^5 - x^4 = x^4(x-1) \). Denominator: \( x^3 - x^2 = x^2(x-1) \). Cancel \( x-1 \) and \( x^2 \) to get \( x^2 \). Integrate \( x^2 \) to get \( \frac{x^3}{3} + c \).
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