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# Solve the trigonometric integral $\int\frac{1}{2\sin\left(x\right)\cos\left(x\right)}dx$

## Step-by-step Solution

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### Videos

$-\frac{1}{2}\ln\left(\cot\left(x\right)\right)+C_0$
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## Step-by-step Solution

Problem to solve:

$\int\frac{1}{2sin\left(x\right)cos\left(x\right)}dx$

Specify the solving method

Simplify $2\sin\left(x\right)\cos\left(x\right)$ using the trig identity: $\sin(2x)=2\sin(x)\cos(x)$

$\int\frac{2}{2\sin\left(2x\right)}dx$

Cancel the fraction's common factor $2$

$\int\frac{1}{\sin\left(2x\right)}dx$

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

$\int\csc\left(2x\right)dx$
1

Simplifying

$\int\csc\left(2x\right)dx$
2

We can solve the integral $\int\csc\left(2x\right)dx$ 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 $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=2x$

Differentiate both sides of the equation $u=2x$

$du=\frac{d}{dx}\left(2x\right)$

Find the derivative

$\frac{d}{dx}\left(2x\right)$

The derivative of the linear function times a constant, is equal to the constant

$2$
3

Now, in order to rewrite $dx$ 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

$du=2dx$
4

Isolate $dx$ in the previous equation

$\frac{du}{2}=dx$
5

Substituting $u$ and $dx$ in the integral and simplify

$\int\frac{\csc\left(u\right)}{2}du$

Take the constant $\frac{1}{2}$ out of the integral

$\frac{1}{2}\int\csc\left(u\right)du$

Divide $1$ by $2$

$\frac{1}{2}\int\csc\left(u\right)du$
6

Take the constant $\frac{1}{2}$ out of the integral

$\frac{1}{2}\int\csc\left(u\right)du$

The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$

$\frac{1}{2}\left(-1\right)\ln\left(\csc\left(u\right)+\cot\left(u\right)\right)$

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

$-\frac{1}{2}\ln\left(\csc\left(u\right)+\cot\left(u\right)\right)$
7

The integral of $\csc(x)$ is $-\ln(\csc(x)+\cot(x))$

$-\frac{1}{2}\ln\left(\csc\left(u\right)+\cot\left(u\right)\right)$

$-\frac{1}{2}\ln\left(\csc\left(2x\right)+\cot\left(2x\right)\right)$
8

Replace $u$ with the value that we assigned to it in the beginning: $2x$

$-\frac{1}{2}\ln\left(\csc\left(2x\right)+\cot\left(2x\right)\right)$

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

$-\frac{1}{2}\ln\left(\frac{1}{\sin\left(2x\right)}+\cot\left(2x\right)\right)$

Apply the trigonometric identity: $\displaystyle\cot(x)=\frac{\cos(x)}{\sin(x)}$

$-\frac{1}{2}\ln\left(\frac{1}{\sin\left(2x\right)}+\frac{\cos\left(2x\right)}{\sin\left(2x\right)}\right)$

Add fraction's numerators with common denominators: $\frac{1}{\sin\left(2x\right)}$ and $\frac{\cos\left(2x\right)}{\sin\left(2x\right)}$

$-\frac{1}{2}\ln\left(\frac{1+\cos\left(2x\right)}{\sin\left(2x\right)}\right)$

Simplify $1+\cos\left(2x\right)$ into $2\cos\left(x\right)^2$ by applying trigonometric identities

$-\frac{1}{2}\ln\left(\frac{2\cos\left(x\right)^2}{\sin\left(2x\right)}\right)$

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

$-\frac{1}{2}\ln\left(\frac{\cos\left(x\right)^2}{\sin\left(x\right)\cos\left(x\right)}\right)$

Simplify the fraction $\frac{\cos\left(x\right)^2}{\sin\left(x\right)\cos\left(x\right)}$ by $\cos\left(x\right)$

$-\frac{1}{2}\ln\left(\frac{\cos\left(x\right)}{\sin\left(x\right)}\right)$

Apply the trigonometric identity: $\frac{\cos\left(x\right)}{\sin\left(x\right)}$$=\cot\left(x\right)$

$-\frac{1}{2}\ln\left(\cot\left(x\right)\right)$
9

Simplify $\csc\left(2x\right)+\cot\left(2x\right)$ using trigonometric identities

$-\frac{1}{2}\ln\left(\cot\left(x\right)\right)$
10

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

$-\frac{1}{2}\ln\left(\cot\left(x\right)\right)+C_0$

$-\frac{1}{2}\ln\left(\cot\left(x\right)\right)+C_0$
SnapXam A2

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1
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5
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7
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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

$\int\frac{1}{2sin\left(x\right)cos\left(x\right)}dx$

### Main topic:

Trigonometric Integrals

~ 0.11 s