![\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](/tpl/images/0989/3146/97a7e.png)
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](/tpl/images/0989/3146/47ba7.png)
Integration Rule 2:
![\int {f(x)+g(x)} \, dx =\int {f(x)} \, dx + \int {g(x)} \, dx](/tpl/images/0989/3146/6e5ae.png)
Integration 1:
![\int {\frac{1}{u} } \, du =ln|u| + C](/tpl/images/0989/3146/5a24b.png)
Integration 2:
![\int {\frac{du}{u^2+a^2} } = \frac{1}{a} arctan(\frac{u}{a} )+C](/tpl/images/0989/3146/19788.png)
Integration 3:
![\int {x^n} \, dx = \frac{x^{n+1}}{n+1} +C](/tpl/images/0989/3146/a5cea.png)
Step-by-step explanation:
Step 1: Define
![\int {\frac{x^2}{x^2+x+3} } \, dx](/tpl/images/0989/3146/83ac4.png)
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}](/tpl/images/0989/3146/902d9.png)
Now we rewrite our Integral: ![\int ({1-\frac{x+3}{x^2+x+3} }) \, dx](/tpl/images/0989/3146/429d7.png)
Step 3: Integrate Pt. 1
Distributive Integral [Int Rule 1]: Â Â Â Â Â Â Â Â Â Â
![\int {1} \, dx - \int {\frac{x+3}{x^2+x+3} } \, dx](/tpl/images/0989/3146/b3245.png)
Integrate 1st Integral [Int 3]: Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \int {\frac{x+3}{x^2+x+3} } \, dx](/tpl/images/0989/3146/37734.png)
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]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} \int {\frac{2(x+3)}{x^2+x+3} } \, dx](/tpl/images/0989/3146/f2966.png)
Distribute 2 [Alg]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} \int {\frac{2x+6}{x^2+x+3} } \, dx](/tpl/images/0989/3146/90a0e.png)
Rewrite Integral [Alg]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} \int {\frac{2x+1+5}{x^2+x+3} } \, dx](/tpl/images/0989/3146/58632.png)
Rewrite Integral [Int Trick 1]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} [\int {\frac{2x+1}{x^2+x+3} } \, dx + \int {\frac{5}{x^2+x+3} } \, dx ]](/tpl/images/0989/3146/5f9f8.png)
(2nd Int) Complete the Square: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![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 ]](/tpl/images/0989/3146/03924.png)
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: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} [\int {\frac{1}{u} } \, du + \int {\frac{5}{z^2 + (\sqrt{\frac{11}{4}})^2} } \, dz ]](/tpl/images/0989/3146/4e171.png)
Rewrite Integral [Int Rule 1]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} [\int {\frac{1}{u} } \, du + 5\int {\frac{dz}{z^2 + (\sqrt{\frac{11}{4}})^2} } ]](/tpl/images/0989/3146/d248e.png)
Integrate 1st Integral [Int 1]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} [ln|u| + 5\int {\frac{dz}{z^2 + (\sqrt{\frac{11}{4}})^2} } ]](/tpl/images/0989/3146/7181e.png)
Integrate 2nd Integral [Int 2]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} [ln|u| + 5(\frac{1}{\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } ) ) ]](/tpl/images/0989/3146/ef3e3.png)
Distribute 5 [Alg]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2} [ln|u| + \frac{5}{\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } ) ]](/tpl/images/0989/3146/5c924.png)
Distribute -1/2 [Alg]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2}ln|u| - \frac{5}{2\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } )](/tpl/images/0989/3146/c3896.png)
Rationalize [Alg]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2}ln|u| - \frac{5\sqrt{11} }{11}arctan(\frac{z}{\sqrt{\frac{11}{4} } } )](/tpl/images/0989/3146/ede59.png)
Resubstitute variables [Alg]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2}ln|x^2+x+3| - \frac{5\sqrt{11} }{11}arctan(\frac{x+\frac{1}{2} }{\sqrt{\frac{11}{4} } } )](/tpl/images/0989/3146/b63c2.png)
Simplify/Rationalize [Alg]: Â Â Â Â Â Â Â Â Â Â
![x - \frac{1}{2}ln|x^2+x+3| - \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} )](/tpl/images/0989/3146/6e32b.png)
Rewrite [Alg]: Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![- \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x](/tpl/images/0989/3146/51471.png)
Integration Constant: Â Â Â Â Â Â Â Â Â Â Â Â Â Â
![- \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x + C](/tpl/images/0989/3146/dda5c.png)
And we have our final answer! Hope this helped you on your Calculus Journey!
![Integrate the following:](/tpl/images/0989/3146/9d99e.jpg)