Key to "Find the definite integrals. ∫12(3x2−1)dx"

jippie771h

Answered question

2021-11-20

Find the definite integrals.

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

Sact1989

Beginner2021-11-21Added 10 answers

Step 1
The given definite integral is:

I = \int_{1}^{2} (\frac{3}{x^{2}} - 1) d x

= \int_{1}^{2} (3 x^{- 2} - 1) d x

= 3 \int_{1}^{2} (x^{- 2}) d x - \int_{1}^{2} 1 d x

Step 2
Apply the power law of integral,

I = 3 {[\frac{x^{- 2 + 1}}{- 2 + 1}]}_{1}^{2} - {[\frac{x^{0 + 1}}{0 + 1}]}_{1}^{2}

= 3 {[- \frac{1}{x}]}_{1}^{2} - {[x]}_{1}^{2}

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

= \frac{3}{2} - 1

= \frac{1}{2}

=0.5
The value of the definite integral is 0.5.

Traidnew

Beginner2021-11-22Added 14 answers

Step 1: 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 2: In this case,

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

. Find its integral.

- \frac{3}{x} - x ∣_{1}^{2}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(2)−F(1):

(- \frac{3}{2} - 2) - (- \frac{3}{1} - 1)

Step 4: Simplify.

\frac{1}{2}

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