Question

Evaluate the integral of 1/y² Cos (1/y) dy 

Answer 1

Evaluate the integral

$\displaystyle\int\dfrac{1}{y^2}\cos \left(\frac{1}{y}\right) dy$

Note that

$d\left(\dfrac{1}{y}\right)=\left(\dfrac{1}{y}\right)'dy=-\dfrac{1}{y^2}dy$

Then

$\displaystyle\int\dfrac{1}{y^2}\cdot\cos \left(\frac{1}{y}\right) dy =$

$\displaystyle\int\cos \left(\frac{1}{y}\right)\cdot\dfrac{1}{y^2} dy =$

$-\displaystyle\int\cos \left(\frac{1}{y}\right)\cdot\left(-\dfrac{1}{y^2}\right) dy =$

$-\displaystyle\int\cos \left(\frac{1}{y}\right)d\left(\dfrac{1}{y}\right) =$

$-\sin \left(\dfrac{1}{y}\right)+C$