ðŸ‘‰ Try now NerdPal! Our new math app on iOS and Android

## Get detailed solutions to your math problems with our Integrals with Radicals step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

Go!
Symbolic mode
Text mode
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

###  Difficult Problems

1

Here, we show you a step-by-step solved example of integrals with radicals. This solution was automatically generated by our smart calculator:

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

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

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

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

Multiply $-1$ times $4$

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

$\int2\cdot 2\sqrt{1-\sin\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
7

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

$\int2\cdot 2\sqrt{\cos\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
8

The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

$2\int\sqrt{\cos\left(\theta \right)^2}\cos\left(\theta \right)d\theta$
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 $\frac{1}{2}$

$2\int\cos\left(\theta \right)^{2\cdot \left(\frac{1}{2}\right)}\cos\left(\theta \right)d\theta$
10

Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)$

$2\int\cos\left(\theta \right)^{\frac{2}{2}}\cos\left(\theta \right)d\theta$
11

Divide $2$ by $2$

$2\int\cos\left(\theta \right)\cos\left(\theta \right)d\theta$
12

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

$2\int\cos\left(\theta \right)^2d\theta$
13

Apply the formula: $\int\cos\left(\theta \right)^2dx$$=\frac{1}{2}\theta +\frac{1}{4}\sin\left(2\theta \right)+C$, where $x=\theta$

$2\left(\frac{1}{2}\theta +\frac{1}{4}\sin\left(2\theta \right)\right)$
14

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

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}\sin\left(2\theta \right)\right)$
15

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

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

Multiply the fraction and term in $2\left(\frac{1}{4}\right)\sin\left(\theta \right)\cos\left(\theta \right)$

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

Multiply $2$ times $1$

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

Divide $2$ by $4$

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{2}\sin\left(\theta \right)\cos\left(\theta \right)\right)$
16

Multiply the fraction and term in $2\left(\frac{1}{4}\right)\sin\left(\theta \right)\cos\left(\theta \right)$

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

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

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{2}\frac{x}{2}\frac{\sqrt{4-x^2}}{2}\right)$

Multiplying fractions $\frac{1}{2} \times \frac{x}{2}$

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{\sqrt{4-x^2}}{2}\frac{1x}{2\cdot 2}\right)$

Multiply $2$ times $2$

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{\sqrt{4-x^2}}{2}\frac{1x}{4}\right)$

Any expression multiplied by $1$ is equal to itself

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{\sqrt{4-x^2}}{2}\frac{x}{4}\right)$

Multiplying fractions $\frac{x}{4} \times \frac{\sqrt{4-x^2}}{2}$

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{4\cdot 2}\right)$

Multiplying fractions $\frac{1}{2} \times \frac{x}{2}$

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{\sqrt{4-x^2}}{2}\frac{1x}{2\cdot 2}\right)$

Multiply $2$ times $2$

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{\sqrt{4-x^2}}{2}\frac{1x}{4}\right)$

Any expression multiplied by $1$ is equal to itself

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{\sqrt{4-x^2}}{2}\frac{x}{4}\right)$

Multiply $4$ times $2$

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{8}\right)$
17

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

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{8}\right)$
18

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

$2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{8}\right)+C_0$

Solve the product $2\left(\frac{1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{8}\right)$

$2\cdot \left(\frac{1}{2}\right)\arcsin\left(\frac{x}{2}\right)+2\left(\frac{x\sqrt{4-x^2}}{8}\right)+C_0$

Multiplying the fraction by $2$

$2\cdot \left(\frac{1}{2}\right)\arcsin\left(\frac{x}{2}\right)+\frac{2x\sqrt{4-x^2}}{8}+C_0$

Take $\frac{2}{8}$ out of the fraction

$2\cdot \left(\frac{1}{2}\right)\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}x\sqrt{4-x^2}+C_0$

Multiply the fraction and term in $2\cdot \left(\frac{1}{2}\right)\arcsin\left(\frac{x}{2}\right)$

$\frac{2\cdot 1}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}x\sqrt{4-x^2}+C_0$

Any expression multiplied by $1$ is equal to itself

$\frac{2}{2}\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}x\sqrt{4-x^2}+C_0$

Divide $2$ by $2$

$\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}x\sqrt{4-x^2}+C_0$
19

Expand and simplify

$\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}x\sqrt{4-x^2}+C_0$

##  Final answer to the problem

$\arcsin\left(\frac{x}{2}\right)+\frac{1}{4}x\sqrt{4-x^2}+C_0$

### Are you struggling with math?

Access detailed step by step solutions to thousands of problems, growing every day!