Question-73030

Question Number 73030 by Tanmay chaudhury last updated on 05/Nov/19

Answered by Tanmay chaudhury last updated on 05/Nov/19

\int_{0}^{4} \frac{l n 2}{l n x} - \frac{l n 2 \times l n 2}{l n x \times l n x \times l n 2} d x

l n 2 \int_{2}^{4} \frac{1}{l n x} - \frac{1}{{(l n x)}^{2}} d x

l n 2 \int_{2}^{4} \frac{l n x - 1}{{(l n x)}^{2}} d x

l n 2 \int_{2}^{4} \frac{l n x . \frac{d x}{d x} - x . \frac{d}{d x} (l n x)}{{(l n x)}^{2}} d \overset{}{x}

l n 2 \int_{2}^{4} \frac{d}{d x} (\frac{x}{l n x}) d x

l n 2 \times ∣ \frac{x}{l n x} ∣_{2}^{4}

l n 2 \times (\frac{4}{l n 4} - \frac{2}{l n 2})

l n 2 \times (\frac{4}{2 l n 2} - \frac{2}{l n 2}) = 0 a n s w e r

Answered by mind is power last updated on 05/Nov/19

\int_{2}^{4} (\log_{x} (2) - \frac{{(\log_{x} (2))}^{2}}{\ln (2)}) dx

= \int \frac{\ln (2)}{\ln (x)} - \frac{\ln (2)}{\ln^{2} (x)} dx

= \ln (2) \int \frac{\ln (x) - 1}{\ln^{2} (x)} dx

= \ln (2) \int d (\frac{x}{\ln (x)}) = \ln 2 {[\frac{x}{\ln (x)}]}_{2}^{4} = \frac{4 \ln (2)}{\ln (4)} - \frac{2 \ln (2)}{\ln (2)} = 2 - 2 = 0

Commented by Tanmay chaudhury last updated on 05/Nov/19

t h a n k y o u s i r \dots

Commented by mind is power last updated on 05/Nov/19

y^{'} re welcom