Evaluate: _{a}^{b} \; x/x dx

Correct Answer is : 12log⁡(ba)⋅log⁡(ba)1/2 (ba) · ≤ft(b/a)21log(ba)⋅log(ab)

Correct Answer is : 12log⁡(ba)⋅log⁡(ba)1/2 (ba) · ≤ft(b/a)21log(ba)⋅log(ab)

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Integral Calculus

Question : 39 of 73

Marks: +1, -0

Evaluate:

\int\limits_{a}^{b} \; \frac{\log x}{x} dx

Solution:

Concept:

Use substitution

u = \log x

to convert the integral into a standard power integral.

Explanation:

Let

I = \int \frac{\log x}{x} \, dx

Set

u = \log x

, then

du = \frac{1}{x} \, dx

Thus

I = \int u \, du = \frac{u^2}{2} = \frac{(\log x)^2}{2}

Evaluate from

a

b

\left[ \frac{(\log x)^2}{2} \right]_a^b = \frac{1}{2} \left[ (\log b)^2 - (\log a)^2 \right]

Factor:

(\log b)^2 - (\log a)^2 = (\log b - \log a)(\log b + \log a) = \log\left(\frac{b}{a}\right) \cdot \log(ba)

Hence the integral equals

\frac{1}{2} \log(ba) \cdot \log\left(\frac{b}{a}\right)

Answer:

Option A:

\frac{1}{2} \log(ba) \cdot \log\left(\frac{b}{a}\right)

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