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
Subtract the values $3$ and $-2$
Divide $1$ by $2$
Any expression to the power of $1$ is equal to that same expression
Simplify the integral of secant 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 $4\left(\frac{\sin\left(\theta\right)\sec\left(\theta\right)^{2}}{2}+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$
Multiply $4$ times $\frac{1}{2}$
Simplify the fraction
Simplifying
The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$
Simplify the fraction
Expressing the result of the integral in terms of the original variable
Add the values $\frac{1}{2}$ and $-1$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Simplify the fraction by $x^2+4$
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration
Expressing the result of the integral in terms of the original variable
Add fraction's numerators with common denominators: $\frac{\sqrt{x^2+4}}{2}$ and $\frac{x}{2}$
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