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 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 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: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
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 $\frac{1}{2}$
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)\sec\left(\theta \right)^2$
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: $\frac{d}{dx}\left(\sec(x)\right)=\sec(x)\cdot\tan(x)\cdot D_x(x)$
The derivative of the linear function is equal to $1$
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: $\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2-1$
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
Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)^2\sec\left(\theta \right)$
Multiply the single term $\sec\left(\theta \right)$ by each term of the polynomial $\left(\sec\left(\theta \right)^2-1\right)$
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
Express the variable $\theta$ in terms of the original variable $x$
Multiplying the fraction by $4\left(\frac{\sqrt{x^2+4}}{2}\right)$
Express the variable $\theta$ in terms of the original variable $x$
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 $-4\left(\frac{\sin\left(\theta \right)\sec\left(\theta \right)^{2}}{3-1}+\frac{3-2}{3-1}\int\sec\left(\theta \right)d\theta\right)$
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 $x$
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 integral $-4\int\sec\left(\theta \right)^{3}d\theta$ results in: $-\frac{1}{2}\sqrt{x^2+4}x-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
Gather the results of all integrals
Combining like terms $\sqrt{x^2+4}x$ and $-\frac{1}{2}\sqrt{x^2+4}x$
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 $x$
The integral $-4\int-\sec\left(\theta \right)d\theta$ results in: $4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
Gather the results of all integrals
Combining like terms $-2\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$ and $4\ln\left(\frac{\sqrt{x^2+4}}{2}+\frac{x}{2}\right)$
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
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