Integral of csc(x/2)

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

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Answer

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

Step by step solution

Problem

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

Solve the integral $\int\csc\left(\frac{x}{2}\right)dx$ applying u-substitution. Let $u$ and $du$ be

$\begin{matrix}u=\frac{x}{2} \\ du=\frac{1}{2}dx\end{matrix}$
2

Isolate $dx$ in the previous equation

$\frac{du}{\frac{1}{2}}=dx$
3

Substituting $u$ and $dx$ in the integral

$\int\frac{\csc\left(u\right)}{\frac{1}{2}}du$
4

Taking the constant out of the integral

$2\int\csc\left(u\right)du$
5

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

$2\left(-1\right)\ln\left(\cot\left(u\right)+\csc\left(u\right)\right)$
6

Substitute $u$ back for it's value, $\frac{x}{2}$

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

Add the constant of integration

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

Answer

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

Problem Analysis

Main topic:

Integration by substitution

Time to solve it:

0.24 seconds

Views:

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