Final Answer
Step-by-step Solution
Specify the solving method
Simplify $\sqrt{\left(1+x^2\right)^3}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $3$ and $n$ equals $\frac{1}{2}$
We can solve the integral $\int\frac{1}{\sqrt{\left(1+x^2\right)^{3}}}dx$ 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
Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
Simplify $\sqrt{\left(\sec\left(\theta \right)^2\right)^{3}}$ 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 $\frac{3}{2}$
Simplify the fraction by $\sec\left(\theta \right)$
Applying the trigonometric identity: $\displaystyle\frac{1}{\sec(\theta)}=\cos(\theta)$
Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$
Express the variable $\theta$ in terms of the original variable $x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$