Solved example of integration by trigonometric substitution
Solve the integral $\int\sqrt{x^2+4}dx$ by trigonometric substitution using the substitution
Substituting in the original integral, we get
Factor by the greatest common divisor $4$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $\tan(x)^2+1=\sec(x)^2$
When multiplying exponents with same base you can add the exponents
Simplifying
The integral of a constant by a function is equal to the constant multiplied by the integral of the function
Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
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 $\frac{x}{2}$
The integral $2\int\sec\left(\theta \right)d\theta$ results in: $2\ln\left|\frac{\sqrt{x^2+4}+x}{2}\right|$
Gather the results of all integrals
Multiplying the fraction by $2$
Multiplying fractions
Simplify the fraction by $x^2+4$
Express the variable $\theta$ in terms of the original variable $x$
Take $\frac{2}{4}$ out of the fraction
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration
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