Final answer to the problem
$\frac{\sqrt{6}}{6}\arctan\left(1.2247475x+1.2247475\right)+C_0$
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Step-by-step Solution
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Intermediate steps
1
Rewrite the expression $\frac{1}{3x^2+6x+5}$ inside the integral in factored form
$\int\frac{1}{3\left(\frac{2}{3}+\left(x+1\right)^2\right)}dx$
Explain this step further
2
Take the constant $\frac{1}{3}$ out of the integral
$\frac{1}{3}\int\frac{1}{\frac{2}{3}+\left(x+1\right)^2}dx$
3
We can solve the integral $\frac{1}{3}\int\frac{1}{\frac{2}{3}+\left(x+1\right)^2}dx$ by applying integration method of trigonometric substitution using the substitution
$x=\frac{\sqrt{6}}{3}\tan\left(\theta \right)-1$
Intermediate steps
4
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
$dx=\frac{\sqrt{6}}{3}\sec\left(\theta \right)^2d\theta$
Explain this step further
5
Substituting in the original integral, we get
$\frac{1}{3}\int\frac{\frac{\sqrt{6}}{3}\sec\left(\theta \right)^2}{\frac{2}{3}+\frac{2}{3}\tan\left(\theta \right)^2}d\theta$
6
Factor the polynomial $\frac{2}{3}+\frac{2}{3}\tan\left(\theta \right)^2$ by it's greatest common factor (GCF): $\frac{2}{3}$
$\frac{1}{3}\int\frac{\frac{\sqrt{6}}{3}\sec\left(\theta \right)^2}{\frac{2}{3}\left(1+\tan\left(\theta \right)^2\right)}d\theta$
7
Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
$\frac{1}{3}\int\frac{\frac{\sqrt{6}}{3}\sec\left(\theta \right)^2}{\frac{2}{3}\sec\left(\theta \right)^2}d\theta$
Why is tan(x)^2+1 = sec(x)^2 ?
8
Taking the constant ($\frac{\sqrt{6}}{3}$) out of the integral
$\frac{2}{3\sqrt{6}}\int\frac{\sec\left(\theta \right)^2}{\frac{2}{3}\sec\left(\theta \right)^2}d\theta$
Intermediate steps
9
Simplify the expression inside the integral
$\frac{2}{3\sqrt{6}}\cdot \int\frac{3}{2}d\theta$
Explain this step further
10
The integral of a constant is equal to the constant times the integral's variable
$\frac{\sqrt{6}}{6}\theta $
Intermediate steps
11
Express the variable $\theta$ in terms of the original variable $x$
$\frac{\sqrt{6}}{6}\arctan\left(1.2247475x+1.2247475\right)$
Explain this step further
12
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$\frac{\sqrt{6}}{6}\arctan\left(1.2247475x+1.2247475\right)+C_0$
Final answer to the problem$\frac{\sqrt{6}}{6}\arctan\left(1.2247475x+1.2247475\right)+C_0$