ln2x-ln4x-dx-x-

Question Number 70483 by Mikaell last updated on 04/Oct/19

\int \frac{l n 2 x}{l n 4 x} . \frac{d x}{x}

Commented by peter frank last updated on 04/Oct/19

t h a n k x

Commented by kaivan.ahmadi last updated on 04/Oct/19

u = l n x \Rightarrow d u = \frac{d x}{x}

l n 2 x = l n 2 + l n x = u + l n 2

l n 4 x = l n 4 + l n x = u + l n 4

\Rightarrow \int \frac{u + l n 2}{u + l n 4} d u = \int \frac{u d u}{u + l n 4} + \int \frac{l n 2 d u}{u + l n 4} =

\int (1 - \frac{l n 4}{u + l n 4}) d u + l n 2 \int \frac{d u}{u + l n 4} =

u - l n 4 l n (u + l n 4) + l n 2 l n (u + l n 4) + C =

u + (l n 2 - l n 4) l n (u + l n 4) + C =

u - l n 2 l n (u + l n 4) + C = l n x - l n 2 l n (l n x + l n 4) + C =

l n x - l n 2 l n (l n 4 x) + C

Commented by Mikaell last updated on 04/Oct/19

t h a n k y o u S i r

Answered by Tanmay chaudhury last updated on 04/Oct/19

l n x = t \to \frac{d t}{d x} = \frac{1}{x}

\int \frac{l n 2 + l n x}{2 l n 2 + l n x} \times \frac{d x}{x}

\int \frac{l n 2 + t}{2 l n 2 + t} \times d t

\frac{1}{4} \int \frac{2 l n 2 + t + t}{2 l n 2 + t} d t

\frac{1}{4} \int d t + \frac{1}{4} \int \frac{2 l n 2 + t - 2 l n 2}{2 l n 2 + t} d t

\frac{1}{2} \int d t - \frac{l n 2}{2} \int \frac{d t}{2 l n 2 + t}

\frac{t}{2} - \frac{l n 2}{2} l n (2 l n 2 + t) + C

\frac{l n x}{2} - \frac{l n 2}{2} l n (2 l n 2 + l n x) + c

Commented by peter frank last updated on 04/Oct/19

t h a n x

Commented by Mikaell last updated on 04/Oct/19

t h a n k y o u S i r

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