zachutnat4o

2021-11-21

$2,1\left(\frac{1}{{x}^{3}}+2\right)$ find the integral

Froldigh

Step 1
To Determine: find the integral of $\left(\frac{1}{{x}^{3}}+2\right)$ in interval (2,1)
Given: we have a function $f\left(x\right)=y=\left(\frac{1}{{x}^{3}}+2\right)$. Also we have limits 2 to 1
Explanation: we have an integral
${\int }_{2}^{1}\left(\frac{1}{{x}^{3}}+2\right)dx$
so we will integrate this as follows
Step 2
${\int }_{2}^{1}\left({x}^{-3}+2\right)dx={\left[\frac{{x}^{-3+1}}{-3+1}+2x\right]}_{1}^{2}$
$={\left[\frac{{x}^{-2}}{-2}+2x\right]}_{2}^{1}$
$={\left[\frac{-1}{2{x}^{2}}+2x\right]}_{2}^{1}$
taking limits we have
${\int }_{2}^{1}\left({x}^{-3}+2\right)dx={\left[\frac{-1}{2{x}^{2}}+2x\right]}_{2}^{1}$
$\left[\left(\frac{-1}{2×{1}^{2}}+2×1\right)-\left(\frac{-1}{2×{2}^{2}}+2×2\right)\right]$
$=\left[\left(\frac{-1}{2}+2\right)-\left(\frac{-1}{8}+4\right)\right]$
$=\left[\frac{-1}{2}+2+\frac{1}{8}-4\right]$
$=\frac{-3}{8}-2$
$=\frac{-19}{8}$

inenge3y

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, $f\left(x\right)=\left(\frac{1}{{x}^{3}}+2\right)$. 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):
$\left(-\frac{1}{2×{1}^{2}}+2×1\right)-\left(-\frac{1}{2×{2}^{2}}+2×2\right)$
Step 4: Simplify.
$-\frac{19}{8}$

Do you have a similar question?