"Evaluate the integral. ∫23[3x2+2x+1x2]dx" solved and explained

druczekq4

Answered question

2021-11-20

Evaluate the integral.

\int_{2}^{3} [3 x^{2} + 2 x + \frac{1}{x^{2}}] d x

breisgaoyz

Beginner2021-11-21Added 14 answers

Step 1
The given integral is

\int_{2}^{3} [3 x^{2} + 2 x + \frac{1}{x^{2}}] d x

.
Step 2
Evaluate the given integral as shown below:

\int_{2}^{3} [3 x^{2} + 2 x + \frac{1}{x^{2}}] d x = \int_{2}^{3} [3 x^{2}] d x + \int_{2}^{3} [2 x] d x + \int_{2}^{3} [\frac{1}{x^{2}}] d x

= 3 \int_{2}^{3} x^{2} d x + 2 \int_{2}^{3} x d x + \int_{2}^{3} x^{- 2} d x

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

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

= [27 - 8] + [9 - 4] - [\frac{1}{3} - \frac{1}{2}]

= [19] + [5] + [\frac{1}{6}]

= \frac{145}{6}

Rosemary McBride

Beginner2021-11-22Added 10 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) = (3 x^{2} + 2 x + \frac{1}{x^{2}})

. Find its integral.

x^{3} + x^{2} - \frac{1}{x} ∣_{2}^{3}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(3)-F(2):

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

Step 4: Simplify.

\frac{145}{6}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?