Step-by-step Solution

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

Go!
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Final Answer

$\frac{1}{2}\ln\left|\tan\left(x\right)\right|+C_0$

Step-by-step explanation

Problem to solve:

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

Choose the solving method

1

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

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

Divide $1$ by $2$

$\frac{1}{2}\int\frac{1}{\sin\left(x\right)\cos\left(x\right)}dx$
3

Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\frac{1}{2}\int\frac{1}{\sin\left(x\right)}\sec\left(x\right)dx$
4

Multiplying the fraction by $\sec\left(x\right)$

$\frac{1}{2}\int\frac{\sec\left(x\right)}{\sin\left(x\right)}dx$
5

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

$u=\tan\left(x\right)$
6

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

$du=\sec\left(x\right)^2dx$
7

Isolate $dx$ in the previous equation

$\frac{du}{\sec\left(x\right)^2}=dx$
8

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

$\frac{1}{2}\int\frac{1}{u}du$
9

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$\frac{1}{2}\ln\left|u\right|$
10

Replace $u$ with the value that we assigned to it in the beginning: $\tan\left(x\right)$

$\frac{1}{2}\ln\left|\tan\left(x\right)\right|$
11

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

$\frac{1}{2}\ln\left|\tan\left(x\right)\right|+C_0$

Final Answer

$\frac{1}{2}\ln\left|\tan\left(x\right)\right|+C_0$
$\int\frac{1}{2sin\left(x\right)cos\left(x\right)}dx$

Time to solve it:

~ 0.36 s (SnapXam)