First slide

Introduction to integration

Question

$\int \frac{e^{6 \log x} - e^{5 \log x}}{e^{4 \log x} - e^{3 \log x}} dx$ is equal to

Easy

A

$\frac{x}{2} + C$
B

$\frac{x^{2}}{2} + C$
C

$\frac{x^{3}}{3} + C$
D

$\frac{x^{4}}{2} + C$

Solution

$\begin{array}{l} \int \frac{e^{6 \log x} - e^{5 \log x}}{e^{4 \log x} - e^{3 \log x}} dx = \int \frac{e^{\log x^{6}} - e^{\log x^{5}}}{e^{\log x^{4}} - e^{\log x^{3}}} dx \\ = \int \frac{x^{6} - x^{5}}{x^{4} - x^{3}} dx \\ = \int \frac{x^{5} (x - 1)}{x^{3} (x - 1)} dx \\ = \int x^{2} dt = \frac{x^{3}}{3} + C \end{array}$ $[∵ \log m^{n} = n \log m]$

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