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Integration by trigonometric substitution Calculator

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1

Example

$\int_{0}^{4}\frac{1}{\sqrt{x\cdot \left(4-x\right)}}dx$
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 $\cos\left(\theta\right)$

$\int_{0}^{4}\frac{8}{2}d\theta$
10

Divide $8$ by $2$

$\int_{0}^{4}4d\theta$
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 $-1$

$arcsin\left(\frac{0}{2}\right)\left(-4\right)+arcsin\left(\frac{4}{2}\right)\cdot 4$
15

Divide $4$ by $2$

$arcsin\left(0\right)\left(-4\right)+arcsin\left(2\right)\cdot 4$
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 $NaN$

$NaN$

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