Step-by-step Solution

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

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Final Answer

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

Problem to solve:

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

Choose 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 $C$

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

Final Answer

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

Tips on how to improve your answer:

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

Related Formulas:

5. See formulas

Time to solve it:

~ 0.18 s