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###  Difficult Problems

1

Solved example of integrals with radicals

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

We can solve the integral $\int\sqrt{4-x^2}dx$ by applying integration method of trigonometric substitution using the substitution

$x=2\sin\left(\theta \right)$

Differentiate both sides of the equation $x=2\sin\left(\theta \right)$

$dx=\frac{d}{d\theta}\left(2\sin\left(\theta \right)\right)$

Find the derivative

$\frac{d}{d\theta}\left(2\sin\left(\theta \right)\right)$

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

$2\frac{d}{d\theta}\left(\sin\left(\theta \right)\right)$

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$2\cos\left(\theta \right)$
3

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

$dx=2\cos\left(\theta \right)d\theta$

4

Substituting in the original integral, we get

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

Factor the polynomial $4-4\sin\left(\theta \right)^2$ by it's greatest common factor (GCF): $4$

$\int2\sqrt{4\left(1-\sin\left(\theta \right)^2\right)}\cos\left(\theta \right)d\theta$
6

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

$\int4\sqrt{1-\sin\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
7

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

$\int4\sqrt{\cos\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
8

The integral of a function times a constant ($4$) is equal to the constant times the integral of the function

$4\int\sqrt{\cos\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
9

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 $\frac{1}{2}$

$4\int\cos\left(\theta \right)\cos\left(\theta \right)d\theta$
10

When multiplying two powers that have the same base ($\cos\left(\theta \right)$), you can add the exponents

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

Apply the trigonometric identity: $\cos\left(x\right)^2$$=\frac{1+\cos\left(2x\right)}{2}, where x=\theta \frac{1+\cos\left(2\theta \right)}{2} 11 Rewrite the trigonometric expression \cos\left(\theta \right)^2 inside the integral 4\int\frac{1+\cos\left(2\theta \right)}{2}d\theta 12 Take the constant \frac{1}{2} out of the integral 4\cdot \left(\frac{1}{2}\right)\int\left(1+\cos\left(2\theta \right)\right)d\theta Divide 1 by 2 4\cdot \frac{1}{2}\int\left(1+\cos\left(2\theta \right)\right)d\theta Expand the integral \int\left(1+\cos\left(2\theta \right)\right)d\theta into 2 integrals using the sum rule for integrals, to then solve each integral separately 4\cdot \frac{1}{2}\left(\int1d\theta+\int\cos\left(2\theta \right)d\theta\right) Multiply 4 times \frac{1}{2} 2\left(\int1d\theta+\int\cos\left(2\theta \right)d\theta\right) 13 Simplify the expression inside the integral 2\left(\int1d\theta+\int\cos\left(2\theta \right)d\theta\right) 14 Solve the product 2\left(\int1d\theta+\int\cos\left(2\theta \right)d\theta\right) 2\cdot \int1d\theta+2\int\cos\left(2\theta \right)d\theta The integral of a constant is equal to the constant times the integral's variable 2\theta Express the variable \theta in terms of the original variable x 2\arcsin\left(\frac{x}{2}\right) 15 The integral 2\cdot \int1d\theta results in: 2\arcsin\left(\frac{x}{2}\right) 2\arcsin\left(\frac{x}{2}\right) Apply the formula: \int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)+C$, where $a=2$ and $x=\theta$

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

Simplify the expression inside the integral

$\sin\left(2\theta \right)$

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

$2\sin\left(\theta \right)\cos\left(\theta \right)$

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

$\frac{\sqrt{4-x^2}x}{2}$
16

The integral $2\int\cos\left(2\theta \right)d\theta$ results in: $\frac{\sqrt{4-x^2}x}{2}$

$\frac{\sqrt{4-x^2}x}{2}$
17

Gather the results of all integrals

$2\arcsin\left(\frac{x}{2}\right)+\frac{\sqrt{4-x^2}x}{2}$
18

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

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

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