Solved example of integrals with radicals
Solve the integral $\int\sqrt{4-x^2}dx$ 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: $1-\sin\left(\theta\right)^2=\cos\left(\theta\right)^2$
Applying the power of a power property
The integral of a constant by a function is equal to the constant multiplied by the integral of the function
Apply the formula: $\cos\left(x\right)^2$$=\frac{1+\cos\left(2x\right)}{2}$, where $x=\theta$
Split the fraction $\frac{1+\cos\left(2\theta\right)}{2}$ inside the integral, in two terms with common denominator $2$
The integral of the sum of two or more functions is equal to the sum of their integrals
The integral of a constant is equal to the constant times the integral's variable
Take the constant out of the integral
Apply the formula: $\int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)$, where $a=2$ and $x=\theta$
Apply the formula: $\int\cos\left(x\right)^2dx$$=\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)$, where $x=\theta$
Using the sine double-angle identity
Simplify the fraction
Expressing the result of the integral in terms of the original variable
As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration
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