AP EAMCET · Maths · Indefinite Integration
\(\int \frac{f(x) g^{\prime}(x)-f^{\prime}(x) g(x)}{f(x) g(x)} \times[\log g(x)-\log f(x)] d x\) is equal to
- A \(\log \left[\frac{g(x)}{f(x)}\right]+C\)
- B \(\frac{1}{2}\left[\log \frac{g(x)}{f(x)}\right]^2+C\)
- C \(\frac{g(x)}{f(x)} \log \left[\frac{g(x)}{f(x)}\right]+C\)
- D \(\log \left[\frac{g(x)}{f(x)}\right]-\frac{g(x)}{f(x)}+C\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{2}\left[\log \frac{g(x)}{f(x)}\right]^2+C\)
Step-by-step Solution
Detailed explanation
Let \(I=\int \frac{f(x) g^{\prime}(x)-f^{\prime}(x) g(x)}{f(x) g(x)}\) \(\begin{aligned} & {[\log g(x)-\log f(x)] d x} \\ & =\int \log \left[\frac{g(x)}{f(x)}\right] \cdot d\left\{\log \frac{g(x)}{f(x)}\right\} \end{aligned}\) Put \(t=\log \left[\frac{g(x)}{f(x)}\right]\)…
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