Solved example of integration by trigonometric substitution
We can solve the integral $\int\sqrt{x^2+4}dx$ by applying integration method of trigonometric substitution using the substitution
Differentiate both sides of the equation $x=2\tan\left(\theta \right)$
Find the derivative
The derivative of a function multiplied by a constant ($2$) 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
Factor the polynomial $4\tan\left(\theta \right)^2+4$ by it's GCF: $4$
The power of a product is equal to the product of it's factors raised to the same power
Calculate the power $2^2$
Calculate the power $\sqrt{4}$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
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 $\frac{1}{2}$
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)\sec\left(\theta \right)^2$
Subtract the values $3$ and $-2$
Any expression to the power of $1$ is equal to that same expression
Rewrite $\sec\left(\theta \right)^{3}$ as the product of two secants
We can solve the integral $\int\sec\left(\theta \right)^2\sec\left(\theta \right)d\theta$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
Taking the derivative of secant function: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$
First, identify $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
The integral of $\sec(x)^2$ is $\tan(x)$
When multiplying two powers that have the same base ($\tan\left(\theta \right)$), you can add the exponents
Now replace the values of $u$, $du$ and $v$ in the last formula
Multiply the single term $4$ by each term of the polynomial $\left(\tan\left(\theta \right)\sec\left(\theta \right)-\int\tan\left(\theta \right)^2\sec\left(\theta \right)d\theta\right)$
Applying the trigonometric identity: $\tan^2(\theta)=\sec(\theta)^2-1$
We identify that the integral has the form $\int\tan^m(x)\sec^n(x)dx$. If $n$ is odd and $m$ is even, then we need to express everything in terms of secant, expand and integrate each function separately
Express the variable $\theta$ in terms of the original variable $x$
Multiplying the fraction by $4\left(\frac{\sqrt{x^2+4}}{2}\right)$
Rewrite the integrand $\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)$ in expanded form
Expand the integral $\int\left(\sec\left(\theta \right)^{3}-\sec\left(\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
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 $x$
Add fraction's numerators with common denominators: $\frac{\sqrt{x^2+4}}{2}$ and $\frac{x}{2}$
Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
Solve the product $-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta \right)d\theta\right)$
Simplify the fraction $-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{2}\right)$
Express the variable $\theta$ in terms of the original variable $x$
Multiplying the fraction by $-2\left(\frac{\sqrt{x^2+4}}{2}\right)^{2}$
Simplify the fraction by $x^2+4$
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 $x$
Simplifying
The integral $-4\int\left(\sec\left(\theta \right)^2-1\right)\sec\left(\theta \right)d\theta$ results in: $-\frac{1}{2}x\sqrt{x^2+4}+2\ln\left(\frac{\sqrt{x^2+4}+x}{2}\right)$
Gather the results of all integrals
Combining like terms $x\sqrt{x^2+4}$ and $-\frac{1}{2}x\sqrt{x^2+4}$
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
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