Mathematics, 16.12.2020 05:30 TheHomieJaay3092

Integrate the following:

$\displaystyle \int \frac{x^2}{x^2+x+3}\, dx$

Answers: 3

Show answers

Answers

Answer from: sanchezr7187

$\int {\frac{x^2}{x^2+x+3} } \, dx = - \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x + C$

General Formulas and Concepts:

Pre-Algebra

Distributive Property

Algebra I

Completing the SquareRearranging Variables

Algebra II

Long Division

Calculus

U-Substitution[Integration Trick 1] Numerator Split[Integration Trick 2] Completing the SquareIntegration Rule 1: $\int {cf(x)} \, dx = c\int {f(x)} \, dx$ Integration Rule 2: $\int {f(x)+g(x)} \, dx =\int {f(x)} \, dx + \int {g(x)} \, dx$ Integration 1: $\int {\frac{1}{u} } \, du =ln|u| + C$ Integration 2: $\int {\frac{du}{u^2+a^2} } = \frac{1}{a} arctan(\frac{u}{a} )+C$ Integration 3: $\int {x^n} \, dx = \frac{x^{n+1}}{n+1} +C$

Step-by-step explanation:

Step 1: Define

$\int {\frac{x^2}{x^2+x+3} } \, dx$

Step 2: Simplify Function

We do long division to simplify the function inside the function.

See Attachment for Long Division Work.

Once we do long division, our function becomes $1-\frac{x+3}{x^2+x+3}$

Now we rewrite our Integral: $\int ({1-\frac{x+3}{x^2+x+3} }) \, dx$

Step 3: Integrate Pt. 1

Distributive Integral [Int Rule 1]: $\int {1} \, dx - \int {\frac{x+3}{x^2+x+3} } \, dx$ Integrate 1st Integral [Int 3]: $x - \int {\frac{x+3}{x^2+x+3} } \, dx$

Step 4: Identify Variables Pt.1

Set variables for u-substitution.

u = x² + x + 3

du = (2x + 1)dx

Step 5: Integrate Pt. 2

Rewrite Integral [Int Rule 1]: $x - \frac{1}{2} \int {\frac{2(x+3)}{x^2+x+3} } \, dx$ Distribute 2 [Alg]: $x - \frac{1}{2} \int {\frac{2x+6}{x^2+x+3} } \, dx$ Rewrite Integral [Alg]: $x - \frac{1}{2} \int {\frac{2x+1+5}{x^2+x+3} } \, dx$ Rewrite Integral [Int Trick 1]: $x - \frac{1}{2} [\int {\frac{2x+1}{x^2+x+3} } \, dx + \int {\frac{5}{x^2+x+3} } \, dx ]$ (2nd Int) Complete the Square: $x - \frac{1}{2} [\int {\frac{2x+1}{x^2+x+3} } \, dx + \int {\frac{5}{(x+\frac{1}{2})^2 + \frac{11}{4} } } \, dx ]$

Step 6: Identify Variables Pt. 2

Set variables for u-substitution for 2nd integral.

z = x + 1/2

dz = dx

a = √(11/4)

Step 7: Integrate Pt. 3

[Integrate] U-Substitution: $x - \frac{1}{2} [\int {\frac{1}{u} } \, du + \int {\frac{5}{z^2 + (\sqrt{\frac{11}{4}})^2} } \, dz ]$ Rewrite Integral [Int Rule 1]: $x - \frac{1}{2} [\int {\frac{1}{u} } \, du + 5\int {\frac{dz}{z^2 + (\sqrt{\frac{11}{4}})^2} } ]$ Integrate 1st Integral [Int 1]: $x - \frac{1}{2} [ln|u| + 5\int {\frac{dz}{z^2 + (\sqrt{\frac{11}{4}})^2} } ]$ Integrate 2nd Integral [Int 2]: $x - \frac{1}{2} [ln|u| + 5(\frac{1}{\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } ) ) ]$ Distribute 5 [Alg]: $x - \frac{1}{2} [ln|u| + \frac{5}{\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } ) ]$ Distribute -1/2 [Alg]: $x - \frac{1}{2}ln|u| - \frac{5}{2\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } )$ Rationalize [Alg]: $x - \frac{1}{2}ln|u| - \frac{5\sqrt{11} }{11}arctan(\frac{z}{\sqrt{\frac{11}{4} } } )$ Resubstitute variables [Alg]: $x - \frac{1}{2}ln|x^2+x+3| - \frac{5\sqrt{11} }{11}arctan(\frac{x+\frac{1}{2} }{\sqrt{\frac{11}{4} } } )$ Simplify/Rationalize [Alg]: $x - \frac{1}{2}ln|x^2+x+3| - \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} )$ Rewrite [Alg]: $- \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x$ Integration Constant: $- \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x + C$

And we have our final answer! Hope this helped you on your Calculus Journey!

Answer from: kayleewoodard

hope this will help you........

Answer from: Quest

10010010001010010101001001010010100101010010101010010010101010110010001011010010010001

step-by-step explanation:

Answer from: Quest

it is 10.6 according to my calculations

What is the value of x? round to the nearest tenth.

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 14:30, lizzie3545

Which quadratic function best fits this data? x y 1 32 2 78 3 178 4 326 5 390 6 337 a. y=11.41x2+154.42x−143.9 b. y=−11.41x2+154.42x−143.9 c. y=11.41x2+154.42x+143.9 d. y=−11.41x2+154.42x+143.9

Answers: 3

continue

Mathematics, 21.06.2019 19:00, GodlyGamer8239

Billy plotted −3 4 and −1 4 on a number line to determine that −3 4 is smaller than −1 4 .is he correct? explain why or why not

Answers: 3

continue

Mathematics, 21.06.2019 21:50, shay68596

What is the next step in the given proof? choose the most logical approach. a. statement: m 1 + m 2 + 2(m 3) = 180° reason: angle addition b. statement: m 1 + m 3 = m 2 + m 3 reason: transitive property of equality c. statement: m 1 = m 2 reason: subtraction property of equality d. statement: m 1 + m 2 = m 2 + m 3 reason: substitution property of equality e. statement: 2(m 1) = m 2 + m 3 reason: substitution property of equality

Answers: 3

continue

Mathematics, 21.06.2019 22:50, Yangster9305

Ur environment is very sensitive to the amount of ozone in the upper atmosphere. the level of ozone normally found is 4.04.0 parts/million (ppm). a researcher believes that the current ozone level is not at a normal level. the mean of 2525 samples is 4.44.4 ppm with a standard deviation of 0.80.8. assume the population is normally distributed. a level of significance of 0.020.02 will be used. make the decision to reject or fail to reject the null hypothesis. answer tables

Answers: 3

continue

You know the right answer?

Integrate the following:
...