Question

Integrate the following:
$\displaystyle \int \frac{x^2}{x^2+x+3}\, dx$

Answer 1

Answer:

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

General Formulas and Concepts:

Pre-Algebra

• Distributive Property

Algebra I

• Completing the Square
• Rearranging Variables

Algebra II

• Long Division

Calculus

• U-Substitution
• [Integration Trick 1] Numerator Split
• [Integration Trick 2] Completing the Square
• Integration 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

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

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

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

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