Evaluate the following integrals. Include absolute values only when needed.

$\int \frac{\mathrm{sin}\left(\mathrm{ln}x\right)}{4x}dx$

Josalynn
2021-10-15
Answered

Evaluate the following integrals. Include absolute values only when needed.

$\int \frac{\mathrm{sin}\left(\mathrm{ln}x\right)}{4x}dx$

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Talisha

Answered 2021-10-16
Author has **93** answers

It is given that, $\int \frac{\mathrm{sin}\left(\mathrm{ln}\left(x\right)\right)}{4x}dx$

We have to evaluate it.

We have,$\int \frac{\mathrm{sin}\left(\mathrm{ln}\left(x\right)\right)}{4x}dx$

Let$u=\mathrm{ln}\left(x\right)$

differentiate equation w.r.t x we get

$du=\frac{1}{x}dx,\Rightarrow dx=xdu$

Then equation becomes

$\Rightarrow \int \frac{\mathrm{sin}\left(\mathrm{ln}\left(x\right)\right)}{4x}dx=\frac{1}{4}\int \mathrm{sin}\left(u\right)du$

$\Rightarrow \int \frac{\mathrm{sin}\left(\mathrm{ln}\left(x\right)\right)}{4x}dx=\frac{1}{4}(-\mathrm{cos}\left(u\right))+C$ , where C is arbitrary constant

Putting the value of$u=\mathrm{ln}\left(x\right)$ in above equation, we get

$\Rightarrow \int \frac{\mathrm{sin}\left(\mathrm{ln}\left(x\right)\right)}{4x}dx=\frac{1}{4}(-\mathrm{cos}\left(u\right))+C$

Hence,$\int \frac{\mathrm{sin}\left(\mathrm{ln}\left(x\right)\right)}{4x}dx=\frac{\mathrm{cos}\left(\mathrm{ln}\left(x\right)\right)}{4}+C$

We have to evaluate it.

We have,

Let

differentiate equation w.r.t x we get

Then equation becomes

Putting the value of

Hence,

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