Answered: "Evaluate the integral. ∫49(x+1x)2dx"

trainart1

Answered question

2021-11-16

Evaluate the integral.

\int_{4}^{9} {(\sqrt{x} + \frac{1}{\sqrt{x}})}^{2} d x

tnie54

Beginner2021-11-17Added 18 answers

Step 1
Definite integral:

I = \int_{4}^{9} {(\sqrt{x} + \frac{1}{\sqrt{x}})}^{2} d x

{(\sqrt{x} + \frac{1}{\sqrt{x}})}^{2}

= x + 2 \cdot \sqrt{x} \cdot \frac{1}{\sqrt{x}} + \frac{1}{x}

= x + 2 + \frac{1}{x}

Step 2
Now we should find the integration

= \int_{4}^{9} (x + 2 + \frac{1}{x}) d x

= {[\frac{x^{2}}{2} + 2 x + \ln x]}_{4}^{9} = \frac{81}{2} + 18 + \ln 9 - \frac{16}{2} - 8 - \ln 2

= \frac{65}{2} + 10 + \ln \frac{9}{2}

= \frac{85}{2} + \ln \frac{9}{2}

John Timms

Beginner2021-11-18Added 8 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) = {(\sqrt{x} + \frac{1}{\sqrt{x}})}^{2}

. Find its integral.

\frac{x^{2}}{2} + 2 x + \ln x ∣_{4}^{9}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(9)-F(4):

(\frac{9^{2}}{2} + 2 \times 9 + \ln 9) - (\frac{4^{2}}{2} + 2 \times 4 + \ln 4)

Step 4: Simplify.

\frac{85}{2} + \ln 9 - \ln 4

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?