## 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!
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
|◻|
=
>
<
>=
<=
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

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 by the greatest common divisor $4$

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

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

Calculate the power $2^2$

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

Multiply $-1$ times $4$

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

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

Calculate the power $\sqrt{4}$

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

Multiply $2$ times $2$

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

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

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

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

Apply the trigonometric identity: $\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 4\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 Multiply 4 times \frac{1}{2} 2\int\left(1+\cos\left(2\theta \right)\right)d\theta 10 Take the constant \frac{1}{2} out of the integral 2\int\left(1+\cos\left(2\theta \right)\right)d\theta 11 Expand the integral \int\left(1+\cos\left(2\theta \right)\right)d\theta 2\left(\int1d\theta+\int\cos\left(2\theta \right)d\theta\right) 2\left(1\theta +\int\cos\left(2\theta \right)d\theta\right) Any expression multiplied by 1 is equal to itself 2\left(\theta +\int\cos\left(2\theta \right)d\theta\right) 12 The integral of a constant is equal to the constant times the integral's variable 2\left(\theta +\int\cos\left(2\theta \right)d\theta\right) 2\left(\theta +\frac{1}{2}\sin\left(2\theta \right)\right) Divide 1 by 2 2\left(\theta +\frac{1}{2}\sin\left(2\theta \right)\right) We can solve the integral \int\cos\left(2\theta \right)d\theta by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that 2\theta it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part u=2\theta Now, in order to rewrite d\theta in terms of du, we need to find the derivative of u. We need to calculate du, we can do that by deriving the equation above du=2d\theta Isolate d\theta in the previous equation \frac{du}{2}=d\theta Substituting u and d\theta in the integral and simplify \int\frac{\cos\left(u\right)}{2}du Take the constant \frac{1}{2} out of the integral \frac{1}{2}\int\cos\left(u\right)du Apply the integral of the cosine function: \int\cos(x)dx=\sin(x) \frac{1}{2}\sin\left(u\right) Replace u with the value that we assigned to it in the beginning: 2\theta \frac{1}{2}\sin\left(2\theta \right) 13 Apply the formula: \int\cos\left(ax\right)dx$$=\frac{1}{a}\sin\left(ax\right)$, where $a=2$ and $x=\theta$

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

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

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

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

Multiply $2$ times $\frac{1}{2}$

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

Any expression multiplied by $1$ is equal to itself

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

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

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

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

$2\arcsin\left(\frac{x}{2}\right)+2\sin\left(\theta \right)\cos\left(\theta \right)$
17

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

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

Multiplying the fraction by $2$

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

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

$2\arcsin\left(\frac{x}{2}\right)+\frac{x\sqrt{4-x^2}}{2}$
20

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

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

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