Explained: "Evaluate the following integral. ∫(5x6−4x)dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Ashley Bell

Answered question

2022-01-02

Evaluate the following integral.

\int (\frac{5}{x^{6}} - 4 \sqrt{x}) d x

Answer & Explanation

Bertha Jordan

Beginner2022-01-03Added 37 answers

Step 1
Given integral is

\int (\frac{5}{x^{6}} - 4 \sqrt{x}) d x

= \int (5 x^{- 6} - 4 x^{\frac{1}{2}}) d x

We will use the following result to evaluate the integral

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + c

Step 2
Therefore, we have

\int (\frac{5}{x^{6}} - 4 \sqrt{x}) d x = 5 (\frac{x^{- 6 + 1}}{- 6 + 1}) - 4 (\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}) + c

\int (\frac{5}{x^{6}} - 4 \sqrt{x}) d x = - x^{- 5} - 4 (\frac{x^{\frac{3}{2}}}{\frac{3}{2}}) + c

\int (\frac{5}{x^{6}} - 4 \sqrt{x}) d x = - \frac{1}{x^{5}} - \frac{8}{3} x^{\frac{3}{2}} + c

Step 3
Ans:

\int (\frac{5}{x^{6}} - 4 \sqrt{x}) d x = - \frac{1}{x^{5}} - \frac{8}{3} x^{\frac{3}{2}} + c

Kindlein6h

Beginner2022-01-04Added 27 answers

Step 1
Given:

\int \frac{5}{x^{6}} - 4 \sqrt{x} d x

Step 2
Solution

\int \frac{5}{x^{6}} - 4 x^{\frac{1}{2}} d x

\int \frac{5}{x^{6}} d x - \int 4 x^{\frac{1}{2}} d x

- \frac{1}{x^{5}} - \frac{8 x \sqrt{x}}{3}

Add C
Answer:

- \frac{1}{x^{5}} - \frac{8 x \sqrt{x}}{3} + C

Vasquez

Expert2022-01-07Added 669 answers

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

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