Ashley Bell

2022-01-02

Evaluate the following integral.
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx$

Bertha Jordan

Step 1
Given integral is
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx$
$=\int \left(5{x}^{-6}-4{x}^{\frac{1}{2}}\right)dx$
We will use the following result to evaluate the integral
$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+c$
Step 2
Therefore, we have
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx=5\left(\frac{{x}^{-6+1}}{-6+1}\right)-4\left(\frac{{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)+c$
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx=-{x}^{-5}-4\left(\frac{{x}^{\frac{3}{2}}}{\frac{3}{2}}\right)+c$
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx=-\frac{1}{{x}^{5}}-\frac{8}{3}{x}^{\frac{3}{2}}+c$
Step 3
Ans:
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx=-\frac{1}{{x}^{5}}-\frac{8}{3}{x}^{\frac{3}{2}}+c$

Kindlein6h

Step 1
Given:
$\int \frac{5}{{x}^{6}}-4\sqrt{x}dx$
Step 2
Solution
$\int \frac{5}{{x}^{6}}-4{x}^{\frac{1}{2}}dx$
$\int \frac{5}{{x}^{6}}dx-\int 4{x}^{\frac{1}{2}}dx$
$-\frac{1}{{x}^{5}}-\frac{8x\sqrt{x}}{3}$
$-\frac{1}{{x}^{5}}-\frac{8x\sqrt{x}}{3}+C$

Vasquez

$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx$
Let's represent the initial integral as a sum of table integrals:
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx=\int -4\sqrt{x}dx+\int \frac{5}{{x}^{6}}dx$
$\int \left(-4\sqrt{x}\right)dx$
This is a table integral:
$\int -4\sqrt{x}dx=-\frac{8\ast {x}^{3/2}}{3}+C$
$\int \frac{5}{{x}^{6}}dx$
This is a table integral:
$\int \frac{5}{{x}^{6}}dx=-\frac{1}{{x}^{5}}+C$
$\int \left(\frac{5}{{x}^{6}}-4\sqrt{x}\right)dx=-\frac{8\ast {x}^{3/2}}{3}-\frac{1}{{x}^{5}}+C$

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