Expert answers to "Evaluate this indefinite integral: (x−x2+2x)dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Marenonigt

Answered question

2021-12-18

Evaluate this indefinite integral:

(\sqrt{x} - \frac{x}{2} + \frac{2}{\sqrt{x}}) d x

Answer & Explanation

usumbiix

Beginner2021-12-19Added 33 answers

Step 1
Consider the provided integral,

\int (\sqrt{x} - \frac{x}{2} + \frac{2}{\sqrt{x}}) d x

Solve the above integral.
Apply the sum rule of the integral,

\int (\sqrt{x} - \frac{x}{2} + \frac{2}{\sqrt{x}}) d x = \int \sqrt{x} d x - \int \frac{x}{2} d x + \int \frac{2}{\sqrt{x}} d x

Apply the power rule of the integral

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

.
Step 2
Simplifying further,

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

= \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} - \frac{1}{2} \frac{x^{1 + 1}}{1 + 1} + 2 \frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + C

= \frac{2}{3} x^{\frac{3}{2}} - \frac{x^{2}}{4} + 4 \sqrt{x} + C

Hence.

recoronarrv

Beginner2021-12-20Added 20 answers

\int (- \frac{x}{2} + \sqrt{x} + \frac{2}{\sqrt{x}}) d x

Lets

nick1337

Expert2021-12-28Added 777 answers

$\int (\sqrt{x} - \frac{x}{2} + \frac{2}{\sqrt{x}}) d x$
We represent the original integral as the sum of the table of integrals:
$\int (\sqrt{x} - \frac{x}{2} + \frac{2}{\sqrt{x}}) * d x = \int \sqrt{x} * d x + \int - \frac{x}{2} * d x + \int \frac{2}{\sqrt{x}} * d x$
$\int \sqrt{x} * d x$
This tabulated integral:
$\int \sqrt{x} * d x = \frac{2 * x^{3 / 2}}{3} + C$
$\int (- \frac{x}{2}) * d x$
It is tabulated integral:
$\int - \frac{x}{2} * d x = - \frac{x^{2}}{4} + C$
$\int \frac{2}{\sqrt{x}} * d x$
It is tabulated integral:
$\int \frac{2}{\sqrt{x}} * d x = 4 \sqrt{x} + C$
$\int (\sqrt{x} - \frac{x}{2} + \frac{2}{\sqrt{x}}) * d x = \frac{2 * x^{3 / 2}}{3} - \frac{x^{2}}{4} + 4 \sqrt{x} + C$

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