 # Evaluate the integral \int\frac{\cos(1-\ln(y))}{y}dy Reeves 2020-11-08 Answered
Evaluate the integral $\int \frac{\mathrm{cos}\left(1-\mathrm{ln}\left(y\right)\right)}{y}dy$
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Step 1
We have to find the integrals:
$\int \frac{\mathrm{cos}\left(1-\mathrm{ln}\left(y\right)\right)}{y}dy$
We will find this integrals by substitution method
Let $t=1-\mathrm{ln}\left(y\right)$
Differentiating both sides with respect to 'y', we get
$t=1-\mathrm{ln}\left(y\right)$
$dt=0-\frac{1}{y}dy$
$-dt=\frac{dy}{y}$
Step 2
Now finding integrals putting above value,
$\int \frac{\mathrm{cos}\left(1-\mathrm{ln}\left(y\right)\right)}{y}dy=\int \mathrm{cos}\left(1-\mathrm{ln}\left(y\right)\right)\frac{dy}{y}$
$=\int \mathrm{cos}tdt$
$=\mathrm{sin}t+c$
Since integration of cosine function is sine.
Now putting $t=1-\mathrm{ln}\left(y\right)$, we get
$=\mathrm{sin}\left(1-\mathrm{ln}\left(y\right)\right)+c$.
Hence, integrals of the given expression is $\mathrm{sin}\left(1-\mathrm{ln}\left(y\right)\right)+c$.