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### Difficult Problems

1

Solved example of integrals with radicals

$\int\sqrt{4-x^2}dx$
2

Solve the integral $\int\sqrt{4-x^2}dx$ by trigonometric substitution using the substitution

$\begin{matrix}x=2\sin\left(\theta \right) \\ dx=2\cos\left(\theta \right)d\theta\end{matrix}$
3

Substituting in the original integral, we get

$\int2\sqrt{4-4\sin\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
4

Factor by the greatest common divisor $4$

$\int2\sqrt{4\left(1-\sin\left(\theta \right)^2\right)}\cos\left(\theta \right)d\theta$
5

The power of a product is equal to the product of it's factors raised to the same power

$\int4\sqrt{1-\sin\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
6

Applying the trigonometric identity: $1-\sin\left(\theta\right)^2=\cos\left(\theta\right)^2$

$\int4\cos\left(\theta \right)^2d\theta$
7

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$4\int\cos\left(\theta \right)^2d\theta$

Apply the formula: $\cos\left(x\right)^2$$=\frac{1+\cos\left(2x\right)}{2}, where x=\theta 4\int\frac{1+\cos\left(2\theta \right)}{2}d\theta Split the fraction \frac{1+\cos\left(2\theta \right)}{2} inside the integral, in two terms with common denominator 2 4\int\left(\frac{1}{2}+\frac{\cos\left(2\theta \right)}{2}\right)d\theta The integral of the sum of two or more functions is equal to the sum of their integrals 4\left(\int\frac{1}{2}d\theta+\int\frac{\cos\left(2\theta \right)}{2}d\theta\right) The integral of a constant is equal to the constant times the integral's variable 4\left(\frac{1}{2}\theta +\int\frac{\cos\left(2\theta \right)}{2}d\theta\right) Take the constant out of the integral 4\left(\frac{1}{2}\theta +\frac{1}{2}\int\cos\left(2\theta \right)d\theta\right) 8 Apply the formula: \int\cos\left(x\right)^2dx$$=\frac{1}{2}x+\frac{1}{4}\sin\left(2x\right)$, where $x=\theta$

$4\left(\frac{1}{2}\theta +\frac{1}{4}\sin\left(2\theta \right)\right)$

Any expression multiplied by $1$ is equal to itself

$4\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}\sin\left(2\theta \right)\right)$
9

Express the variable $\theta$ in terms of the original variable $x$

$4\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}\sin\left(2\theta \right)\right)$
10

Solve the product $4\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}\sin\left(2\theta \right)\right)$

$2\arcsin\left(\frac{x}{2}\right)+\sin\left(2\theta \right)$
11

Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$

$2\arcsin\left(\frac{x}{2}\right)+2\sin\left(\theta \right)\cos\left(\theta \right)$

Multiplying the fraction by $2$

$2\arcsin\left(\frac{x}{2}\right)+\frac{2x}{2}\frac{\sqrt{4-x^2}}{2}$

Take $\frac{2}{2}$ out of the fraction

$2\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{2}$
12

Express the variable $\theta$ in terms of the original variable $x$

$2\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{2}$
13

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration

$2\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{2}+C_0$

$2\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{2}+C_0$