Solved example of integrals with radicals
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
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
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 ${f'(x) = \cos(x)\cdot D_x(x)}$
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
Substituting in the original integral, we get
Factor the polynomial $4-4\sin\left(\theta \right)^2$ by it's greatest common factor (GCF): $4$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $1-\sin\left(\theta \right)^2 = \cos\left(\theta \right)^2$
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
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 $\frac{1}{2}$
When multiplying two powers that have the same base ($\cos\left(\theta \right)$), you can add the exponents
Apply the trigonometric identity: $\cos\left(\theta \right)^2$$=\frac{1+\cos\left(2\theta \right)}{2}$, where $x=\theta $
Rewrite the trigonometric expression $\cos\left(\theta \right)^2$ inside the integral
Take the constant $\frac{1}{2}$ out of the integral
Divide $1$ by $2$
Expand the integral $\int\left(1+\cos\left(2\theta \right)\right)d\theta$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Multiply $4$ times $\frac{1}{2}$
Simplify the expression inside the integral
Solve the product $2\left(\int1d\theta+\int\cos\left(2\theta \right)d\theta\right)$
The integral of a constant is equal to the constant times the integral's variable
Express the variable $\theta$ in terms of the original variable $x$
The integral $2\cdot \int1d\theta$ results in: $2\arcsin\left(\frac{x}{2}\right)$
Apply the formula: $\int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)+C$, where $a=2$ and $x=\theta $
Simplify the expression inside the integral
Using the sine double-angle identity: $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$
Express the variable $\theta$ in terms of the original variable $x$
The integral $2\int\cos\left(2\theta \right)d\theta$ results in: $\frac{\sqrt{4-x^2}x}{2}$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Access detailed step by step solutions to thousands of problems, growing every day!