Evaluate the integral of cos2ydy

Question

Elizabeth Witte · Accepted Answer

Step 1
Integration is summation of discrete data. The integral is calculated for the functions to find their area, displacement, volume, that occurs due to combination of small data.
Integration is of two types definite integral and indefinite integral. Indefinite integral are defined where upper and lower limits are not given, whereas in definite integral both upper and lower limit are there.
Step 2
The given integrand is

\int \cos^{2} y d y

. Here, using trigonometric identities

\cos^{2} y

can be written as

\frac{1 + \cos 2 y}{2}

.
Substitute the new identity and integrate

\int \cos^{2} y d y = \int \frac{1 + \cos 2 y}{2} d y

= \frac{1}{2} \int 1 + \cos 2 y d y

= \frac{1}{2} [\int 1 d y + \int \cos 2 y d y]

= \frac{1}{2} [y + \frac{\sin 2 y}{2}] + C

Therefore, value of

\int \cos^{2} y d y

is equal to

\frac{1}{2} [y + \frac{\sin 2 y}{2}] + C

navratitehf · Accepted Answer

Step 1: Use Pythagorean Identities: cos2x=12+cos⁡2x2.  ∫12+cos⁡2y2dy  Step 2: Use Sum Rule: ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx.  ∫12dy+∫cos⁡2y2dy  Step 3: Use this rule: ∫adx=ax+C.  y2+∫cos⁡2y2dy  Step 4: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx.  y2+12∫cos⁡2ydy  Step 5: Use Integration by Substitution on ∫cos⁡2ydy.  Let u=2y, du=2dy, then dy=12du  Step 6: Using u and du above, rewrite ∫cos⁡2ydy.  ∫cos⁡u2du  Step 7: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx.  12∫cos⁡udu  Step 8: Use Trigonometric Integration: the integral of cos⁡u is sin⁡u.  sin⁡u2  Step 9: Substitute u=2y back into the original integral.  sin⁡2y2  Step 10: Rewrite the integral with the completed substitution.  y2+sin⁡2y4  Step 11: Add constant.  y2+sin⁡2y4+C

Evaluate the integral of \cos^{2} ydy

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