zachutnat4o

2021-11-21

Froldigh

Beginner2021-11-22Added 17 answers

Step 1

To Determine: find the integral of$(\frac{1}{{x}^{3}}+2)$ in interval (2,1)

Given: we have a function$f\left(x\right)=y=(\frac{1}{{x}^{3}}+2)$ . Also we have limits 2 to 1

Explanation: we have an integral

${\int}_{2}^{1}(\frac{1}{{x}^{3}}+2)dx$

so we will integrate this as follows

Step 2

$\int}_{2}^{1}({x}^{-3}+2)dx={[\frac{{x}^{-3+1}}{-3+1}+2x]}_{1}^{2$

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

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

taking limits we have

$\int}_{2}^{1}({x}^{-3}+2)dx={[\frac{-1}{2{x}^{2}}+2x]}_{2}^{1$

$[(\frac{-1}{2\times {1}^{2}}+2\times 1)-(\frac{-1}{2\times {2}^{2}}+2\times 2)]$

$=[(\frac{-1}{2}+2)-(\frac{-1}{8}+4)]$

$=[\frac{-1}{2}+2+\frac{1}{8}-4]$

$=\frac{-3}{8}-2$

$=\frac{-19}{8}$

To Determine: find the integral of

Given: we have a function

Explanation: we have an integral

so we will integrate this as follows

Step 2

taking limits we have

inenge3y

Beginner2021-11-23Added 20 answers

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

${\int}_{a}^{b}f\left(x\right)dx=F\left(x\right){\mid}_{a}^{b}=F\left(b\right)-F\left(a\right)$

Step 2: In this case,$f\left(x\right)=(\frac{1}{{x}^{3}}+2)$ . Find its integral.

$-\frac{1}{2{x}^{2}}+2x{\mid}_{2}^{1}$

Step 3: Since$F\left(x\right){\mid}_{a}^{b}=F\left(b\right)-F\left(a\right)$ , expand the above into F(1)-F(2):

$(-\frac{1}{2\times {1}^{2}}+2\times 1)-(-\frac{1}{2\times {2}^{2}}+2\times 2)$

Step 4: Simplify.

$-\frac{19}{8}$

Step 2: In this case,

Step 3: Since

Step 4: Simplify.

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