Here, we show you a step-by-step solved example of integrals with radicals. This solution was automatically generated by our smart calculator:
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 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
The power of a product is equal to the product of it's factors raised to the same power
Multiply $-1$ times $4$
Substituting in the original integral, we get
Factor the polynomial $4-4\sin\left(\theta \right)^2$ by it's greatest common factor (GCF): $4$
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 ($2$) is equal to the constant times the integral of the function
Simplify $\sqrt{\cos\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$
When multiplying two powers that have the same base ($\cos\left(\theta \right)$), you can add the exponents
Apply the formula: $\int\cos\left(\theta \right)^2dx$$=\frac{1}{2}\theta +\frac{1}{4}\sin\left(2\theta \right)+C$, where $x=\theta $
Express the variable $\theta$ in terms of the original variable $x$
Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$
Multiply the fraction and term in $2\left(\frac{1}{4}\right)\sin\left(\theta \right)\cos\left(\theta \right)$
Multiplying fractions $\frac{1}{2} \times \frac{x}{2}$
Multiplying fractions $\frac{x}{4} \times \frac{\sqrt{4-x^2}}{2}$
Express the variable $\theta$ in terms of the original variable $x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the product $2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{8}\right)$
Multiplying the fraction by $2$
Take $\frac{2}{8}$ out of the fraction
Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)\arcsin\left(\frac{x}{2}\right)$
Any expression multiplied by $1$ is equal to itself
Divide $2$ by $2$
Expand and simplify
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