Final Answer
Step-by-step Solution
Specify the solving method
We can solve the integral $\int\frac{1}{\sqrt{4+x^2}}dx$ by applying integration method of trigonometric substitution using the substitution
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
Substituting in the original integral, we get
Factor the polynomial $4+4\tan\left(\theta \right)^2$ by it's greatest common factor (GCF): $4$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
Taking the constant ($2$) out of the integral
Simplify $\sqrt{\sec\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$
Simplify the fraction $\frac{\sec\left(\theta \right)^2}{2\sec\left(\theta \right)}$ by $\sec\left(\theta \right)$
Take the constant $\frac{1}{2}$ out of the integral
Simplify the expression inside the integral
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Express the variable $\theta$ in terms of the original variable $x$
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Combine and simplify all terms in the same fraction with common denominator $2$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Simplify the expression by applying logarithm properties