Solved example of integrals with radicals
We can solve the integral $\int\sqrt{4-x^2}dx$ by applying integration method of trigonometric substitution using the substitution
Differentiate both sides of the equation $x=2\sin\left(\theta \right)$
Find the derivative
The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function
The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$
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 $4-4\sin\left(\theta \right)^2$ by it's GCF: $4$
The power of a product is equal to the product of it's factors raised to the same power
Calculate the power $2^2$
Calculate the power $\sqrt{4}$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $1-\sin\left(\theta\right)^2=\cos\left(\theta\right)^2$
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
Apply the trigonometric identity: $\cos\left(x\right)^2$$=\frac{1+\cos\left(2x\right)}{2}$, where $x=\theta $
Rewrite the trigonometric expression $\cos\left(\theta \right)^2$ inside the integral
Take the constant $\frac{1}{2}$ out of the integral
Divide $1$ by $2$
Take the constant $\frac{1}{2}$ out of the integral
Multiply $4$ times $\frac{1}{2}$
Expand the integral $\int\left(1+\cos\left(2\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Simplifying
The integral of a constant is equal to the constant times the integral's variable
Express the variable $\theta$ in terms of the original variable $x$
The integral $2\int1d\theta$ results in: $2\arcsin\left(\frac{x}{2}\right)$
Apply the formula: $\int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)+C$, where $a=2$ and $x=\theta $
Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$
Express the variable $\theta$ in terms of the original variable $x$
Multiplying the fraction by $2\left(\frac{\sqrt{4-x^2}}{2}\right)$
The integral $2\int\cos\left(2\theta \right)d\theta$ results in: $\frac{1}{2}x\sqrt{4-x^2}$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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