Final Answer
Step-by-step Solution
Specify the solving method
We can solve the integral $\int\frac{16}{x^2\sqrt{x^2+9}}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x^2+9$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $dx$ in the previous equation
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
We can solve the integral $\int\frac{8}{\sqrt{\left(u-9\right)^{3}}\sqrt{u}}du$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $v$), which when substituted makes the integral easier. We see that $\sqrt{u}$ it's a good candidate for substitution. Let's define a variable $v$ and assign it to the choosen part
Now, in order to rewrite $du$ in terms of $dv$, we need to find the derivative of $v$. We need to calculate $dv$, we can do that by deriving the equation above
Isolate $du$ in the previous equation
Rewriting $u$ in terms of $v$
Substituting $v$, $du$ and $u$ in the integral and simplify
We can solve the integral $\int\frac{16}{\sqrt{\left(v^{2}-9\right)^{3}}}dv$ by applying integration method of trigonometric substitution using the substitution
Now, in order to rewrite $d\theta$ in terms of $dv$, we need to find the derivative of $v$. We need to calculate $dv$, we can do that by deriving the equation above
Substituting in the original integral, we get
Factor the polynomial $9\sec\left(\theta \right)^{2}-9$ by it's greatest common factor (GCF): $9$
The power of a product is equal to the product of it's factors raised to the same power
Apply the trigonometric identity: $\sec\left(\theta \right)^2-1$$=\tan\left(\theta \right)^2$, where $x=\theta $
Taking the constant ($48$) out of the integral
Simplify $\sqrt{\left(\tan\left(\theta \right)^2\right)^{3}}$ 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 $\tan\left(\theta \right)$
Take the constant $\frac{1}{27}$ out of the integral
Simplify the expression inside the integral
We can solve the integral $\int\frac{\cos\left(\theta \right)}{\sin\left(\theta \right)^{2}}d\theta$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $w$), which when substituted makes the integral easier. We see that $\sin\left(\theta \right)$ it's a good candidate for substitution. Let's define a variable $w$ and assign it to the choosen part
Now, in order to rewrite $d\theta$ in terms of $dw$, we need to find the derivative of $w$. We need to calculate $dw$, we can do that by deriving the equation above
Isolate $d\theta$ in the previous equation
Substituting $w$ and $d\theta$ in the integral and simplify
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$
Replace $w$ with the value that we assigned to it in the beginning: $\sin\left(\theta \right)$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Express the variable $\theta$ in terms of the original variable $x$
Replace $v$ with the value that we assigned to it in the beginning: $\sqrt{u}$
Replace $u$ with the value that we assigned to it in the beginning: $x^2+9$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$