Integration by trigonometric substitution Calculator
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$\int_{0}^{4}\frac{1}{\sqrt{x\cdot \left(4-x\right)}}dx$
2
The power of a product is equal to the product of it's factors raised to the same power
$\int_{0}^{4}\frac{1}{\sqrt{4-x}\sqrt{x}}dx$
3
Solve the integral $\int\frac{1}{\sqrt{4-x}\sqrt{x}}dx$ by trigonometric substitution using the substitution
$\begin{matrix}x=4\sin\left(\theta\right)^{2} \\ dx=8\cos\left(\theta\right)\sin\left(\theta\right)d\theta\end{matrix}$
4
Substituting in the original integral, we get
$\int_{0}^{4}\frac{8\cos\left(\theta\right)\sin\left(\theta\right)}{2\sqrt{4-4\sin\left(\theta\right)^{2}}\sin\left(\theta\right)}d\theta$
5
Factor by the greatest common divisor $4$
$\int_{0}^{4}\frac{8\cos\left(\theta\right)\sin\left(\theta\right)}{2\sqrt{1-\sin\left(\theta\right)^{2}}\sin\left(\theta\right)}d\theta$
6
Simplifying the fraction by $\sin\left(\theta\right)$
$\int_{0}^{4}\frac{8\cos\left(\theta\right)}{2\sqrt{1-\sin\left(\theta\right)^{2}}}d\theta$
7
Applying the trigonometric identity: $1-\sin\left(\theta\right)^2=\cos\left(\theta\right)^2$
$\int_{0}^{4}\frac{8\cos\left(\theta\right)}{2\sqrt{\cos\left(\theta\right)^2}}d\theta$
8
Applying the power of a power property
$\int_{0}^{4}\frac{8\cos\left(\theta\right)}{2\cos\left(\theta\right)}d\theta$
9
Simplifying the fraction by $\cos\left(\theta\right)$
$\int_{0}^{4}\frac{8}{2}d\theta$
11
The integral of a constant is equal to the constant times the integral's variable
$\left[4\theta\right]_{0}^{4}$
12
Expressing the result of the integral in terms of the original variable
$\left[4arcsin\left(\frac{x}{2}\right)\right]_{0}^{4}$
13
Evaluate the definite integral
$arcsin\left(\frac{4}{2}\right)\cdot 4-1\cdot arcsin\left(\frac{0}{2}\right)\cdot 4$
14
Multiply $4$ times $-1$
$arcsin\left(\frac{0}{2}\right)\left(-4\right)+arcsin\left(\frac{4}{2}\right)\cdot 4$
$arcsin\left(0\right)\left(-4\right)+arcsin\left(2\right)\cdot 4$
16
Calculating the arcsine of $2$
$0\left(-4\right)+NaN\cdot 4$
17
Any expression multiplied by $0$ is equal to $0$
$0+NaN\cdot 4$
18
Multiply $4$ times $NaN$
$0+NaN$
19
Subtract the values $0$ and $NaN$
$NaN$