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 method of trigonometric substitution using the substitution
Differentiate both sides of the equation $x=3\tan\left(\theta \right)$
Find the derivative
The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function
The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$
The derivative of the linear function is equal to $1$
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
Rewrite the fraction $\frac{\frac{16}{3}\sec\left(\theta \right)^2}{\tan\left(\theta \right)^2\sqrt{9\tan\left(\theta \right)^2+9}}$
Multiplying the fraction by $\sec\left(\theta \right)^2$
Simplifying
Factor the polynomial $9\tan\left(\theta \right)^2+9$ 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
Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
Taking the constant ($16$) 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 $\frac{\sec\left(\theta \right)^2}{9\tan\left(\theta \right)^2\sec\left(\theta \right)}$ by $\sec\left(\theta \right)$
Take the constant $\frac{1}{9}$ out of the integral
Multiply $16$ times $\frac{1}{9}$
Apply the trigonometric identity: $\frac{\sec\left(\theta \right)}{\tan\left(\theta \right)^m}$$=\frac{\cos\left(\theta \right)^{\left(m-1\right)}}{\sin\left(\theta \right)^m}$, where $x=\theta $ and $m=2$
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 $u$), 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 $u$ and assign it to the choosen part
Differentiate both sides of the equation $u=\sin\left(\theta \right)$
Find the derivative
The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$
Now, in order to rewrite $d\theta$ 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 $d\theta$ in the previous equation
Simplify the fraction $\frac{\frac{\cos\left(\theta \right)}{u^2}}{\cos\left(\theta \right)}$ by $\cos\left(\theta \right)$
Substituting $u$ 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 $u$ 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 $x$
Multiplying the fraction by $9$
Divide fractions $\frac{-16}{\frac{9x}{\sqrt{x^2+9}}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
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 $C$