Evaluate the integral. ∫cos⁡(1−ln⁡(y))ydy

Question

Evaluate the integral.  ∫cos⁡(1−ln⁡(y))ydy

huckelig75 · Accepted Answer

Step 1  We have to find the integrals:  ∫cos⁡(1−ln⁡(y))ydy  We will find this integrals by substitution method  Let t=1−ln⁡(y)  Differentiating both sides with respect to y, we get  t=1−ln⁡(y)  dt=0−1ydy  −dt=dyy  Step 2  Now finding integrals putting above value,  ∫cos⁡(1−ln⁡(y))ydy=∫cos⁡(1−ln⁡(y))dyy  =∫cos⁡tdt  =sin⁡t+c.  Since integration of cosine function is sine.  Now putting t=1−ln⁡(y), we get  =sin⁡(1−ln⁡(y))+c.  Hence, integrals of the given expression is sin⁡(1−ln⁡(y))+c.

Provere · Accepted Answer

Step 1: Use Integration by Substitution.
Let

u = 1 - \ln y, d u = - \frac{1}{y} d y

Step 2: Using u and du above, rewrite

\int \frac{\cos (1 - \ln (y))}{y} d y

.

\int - \cos u d u

Step 3: Use Trigonometric Integration: the integral of

\cos u i s \sin u

.

- \sin u

Step 4: Substitute

u = 1 - \ln y

back into the original integral.

- \sin (1 - \ln y)

Step 5: Add constant.

- \sin (1 - \ln y) + C

Evaluate the integral. \int \frac{\cos (1-\ln (y))}{y}dy

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