Here, we show you a step-by-step solved example of intégrales avec radicaux. 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
Apply the formula: $\frac{d}{dx}\left(cx\right)$$=c\frac{d}{dx}\left(x\right)$
Apply the trigonometric identity: $\frac{d}{dx}\left(\sin\left(\theta \right)\right)$$=\cos\left(\theta \right)$, where $x=\theta $
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
Apply the formula: $\left(ab\right)^n$$=a^nb^n$, where $a=2$, $b=\sin\left(\theta \right)$ and $n=2$
Apply the formula: $ab$$=ab$, where $ab=- 4\sin\left(\theta \right)^2$, $a=-1$ and $b=4$
Substituting in the original integral, we get
Factor the polynomial $4-4\sin\left(\theta \right)^2$ by it's greatest common factor (GCF): $4$
Apply the formula: $\left(ab\right)^n$$=a^nb^n$, where $a=1-\sin\left(\theta \right)^2$, $b=4$ and $n=\frac{1}{2}$
Applying the trigonometric identity: $1-\sin\left(\theta \right)^2 = \cos\left(\theta \right)^2$
Apply the formula: $\int cxdx$$=c\int xdx$, where $c=2$ and $x=2\sqrt{\cos\left(\theta \right)^2}\cos\left(\theta \right)$
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 $\frac{1}{2}$
Apply the formula: $x\cdot x$$=x^2$, where $x=\cos\left(\theta \right)$
Apply the formula: $\int\cos\left(\theta \right)^2dx$$=\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 $x$
Apply the trigonometric identity: $\sin\left(2\theta \right)$$=2\sin\left(\theta \right)\cos\left(\theta \right)$, where $x=\theta $
Apply the formula: $\frac{a}{b}c$$=\frac{ca}{b}$, where $a=1$, $b=4$, $c=2$, $a/b=\frac{1}{4}$ and $ca/b=2\left(\frac{1}{4}\right)\sin\left(\theta \right)\cos\left(\theta \right)$
Apply the formula: $\frac{a}{b}\frac{c}{f}$$=\frac{ac}{bf}$, where $a=1$, $b=2$, $c=x$, $a/b=\frac{1}{2}$, $f=2$, $c/f=\frac{x}{2}$ and $a/bc/f=\frac{1}{2}\frac{x}{2}\frac{\sqrt{4-x^2}}{2}$
Apply the formula: $\frac{a}{b}\frac{c}{f}$$=\frac{ac}{bf}$, where $a=x$, $b=4$, $c=\sqrt{4-x^2}$, $a/b=\frac{x}{4}$, $f=2$, $c/f=\frac{\sqrt{4-x^2}}{2}$ and $a/bc/f=\frac{x}{4}\frac{\sqrt{4-x^2}}{2}$
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 $C$
Apply the formula: $x\left(a+b\right)$$=xa+xb$, where $a=\frac{1}{2}\arcsin\left(\frac{x}{2}\right)$, $b=\frac{x\sqrt{4-x^2}}{8}$, $x=2$ and $a+b=\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{8}$
Apply the formula: $a\frac{b}{c}$$=\frac{ba}{c}$, where $a=2$, $b=x\sqrt{4-x^2}$ and $c=8$
Apply the formula: $\frac{ab}{c}$$=\frac{a}{c}b$, where $ab=2x\sqrt{4-x^2}$, $a=2$, $b=x\sqrt{4-x^2}$, $c=8$ and $ab/c=\frac{2x\sqrt{4-x^2}}{8}$
Apply the formula: $\frac{a}{b}c$$=\frac{ca}{b}$, where $a=1$, $b=2$, $c=2$, $a/b=\frac{1}{2}$ and $ca/b=2\cdot \left(\frac{1}{2}\right)\arcsin\left(\frac{x}{2}\right)$
Apply the formula: $1x$$=x$, where $x=2$
Apply the formula: $\frac{a}{b}$$=\frac{a}{b}$, where $a=2$, $b=2$ and $a/b=\frac{2}{2}$
Expand and simplify
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