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

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

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: $\tan(x)^2+1=\sec(x)^2$

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

When multiplying exponents with same base you can add the exponents

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 $4\left(\frac{\sec\left(\theta\right)^{2}\sin\left(\theta\right)}{2}+\frac{1}{2}\int\sec\left(\theta\right)d\theta\right)$

Simplify the fraction

Expressing the result of the integral in terms of the original variable

Simplifying 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 $\frac{x}{2}$