Find out the answer to "Solve the definite integral. ∫12(3y2+y−1y2)"

gamomaniea1

Answered question

2021-11-20

Solve the definite integral.

\int_{1}^{2} (\frac{3 y^{2} + y - 1}{y^{2}})

Steven Arredondo

Beginner2021-11-21Added 18 answers

Step 1
We have to solve the given definite integral:

\int_{1}^{2} (\frac{3 y^{2} + y - 1}{y^{2}}) d y

Rewriting the integral and solving the given integral,

\int_{1}^{2} (\frac{3 y^{2} + y - 1}{y^{2}}) d y = \int_{1}^{2} (\frac{3 y^{2}}{y^{2}} + \frac{y}{y^{2}} - \frac{1}{y^{2}}) d y

= \int_{1}^{2} (3 + \frac{1}{y} - y^{- 2}) d y

= {[3 y + \ln (y) - \frac{y^{- 2 + 1}}{- 2 + 1}]}_{1}^{2}

= {[3 y + \ln (y) + y^{- 1}]}_{1}^{2}

= {[3 y + \ln (y) + \frac{1}{y}]}_{1}^{2}

= [3 \times 2 + \ln (2) + \frac{1}{2} - 3 \times 1 - \ln (1) - 1]

= [6 + \ln (2) + \frac{1}{2} - 3 - 0 - 1]

= [2 + \frac{1}{2} + \ln (2)]

= \frac{5}{2} + \ln (2)

Step 2
Hence, value of given definite integral is

\frac{5}{2} + \ln (2)

Alicia Washington

Beginner2021-11-22Added 23 answers

Step 1: Simplify

\frac{3 y^{2} + y - 1}{y^{2}} \to \frac{1}{y} + \frac{3 y^{2} - 1}{y^{2}}

\int_{1}^{2} \frac{1}{y} + \frac{3 y^{2} - 1}{y^{2}} d y

Step 2: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

\int_{a}^{b} f (x) d x = F (x) ∣_{a}^{b} = F (b) - F (a)

Step 3: In this case,

f (y) = \frac{1}{y} + \frac{3 y^{2} - 1}{y^{2}}

. Find its integral.

\ln y + 3 y + \frac{1}{y} ∣_{1}^{2}

Step 4: Since

F (y) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(2)−F(1):

(\ln 2 + 3 \times 2 + \frac{1}{2}) - (\ln 1 + 3 \times 1 + \frac{1}{1})

Step 5: Simplify.

\frac{5}{2} + \ln 2

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