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 by the greatest common divisor $4$
Calculate the power $2^2$
Multiply $-1$ times $4$
Calculate the power $\sqrt{4}$
Multiply $2$ times $2$
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 constant by a function is equal to the constant multiplied by the integral of the function
Apply the trigonometric identity: $\cos\left(x\right)^2$$=\frac{1+\cos\left(2x\right)}{2}$, where $x=\theta $
Divide $1$ by $2$
Multiply $4$ times $\frac{1}{2}$
Take the constant $\frac{1}{2}$ out of the integral
Expand the integral $\int\left(1+\cos\left(2\theta \right)\right)d\theta$
Any expression multiplied by $1$ is equal to itself
The integral of a constant is equal to the constant times the integral's variable
Divide $1$ by $2$
We can solve the integral $\int\cos\left(2\theta \right)d\theta$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2\theta $ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $d\theta$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Isolate $d\theta$ in the previous equation
Substituting $u$ and $d\theta$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
Apply the integral of the cosine function: $\int\cos(x)dx=\sin(x)$
Replace $u$ with the value that we assigned to it in the beginning: $2\theta $
Apply the formula: $\int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)$, where $a=2$ and $x=\theta $
Express the variable $\theta$ in terms of the original variable $x$
Multiply $2$ times $\frac{1}{2}$
Any expression multiplied by $1$ is equal to itself
Solve the product $2\left(\arcsin\left(\frac{x}{2}\right)+\frac{1}{2}\sin\left(2\theta \right)\right)$
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 $2$
Take $\frac{2}{2}$ 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 $C$
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