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Integral of ctg(x)-0 dx

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 pi          
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4

$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot{\left(x \right)}\, dx$$

Integral(cot(x), (x, pi/4, pi/2))

Detail solution

Rewrite the integrand:

$\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Let $u = \sin{\left(x \right)}$ .

Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :

$\int \frac{1}{u}\, du$
1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
Now substitute $u$ back in:

$\log{\left(\sin{\left(x \right)} \right)}$
Add the constant of integration:

$\log{\left(\sin{\left(x \right)} \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(\sin{\left(x \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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 | cot(x) dx = C + log(sin(x))
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$$\int \cot{\left(x \right)}\, dx = C + \log{\left(\sin{\left(x \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    /  ___\
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-log|-----|
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$$- \log{\left(\frac{\sqrt{2}}{2} \right)}$$

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-log|-----|
    \  2  /

$$- \log{\left(\frac{\sqrt{2}}{2} \right)}$$

-log(sqrt(2)/2)

Numerical answer [src]

0.346573590279973

0.346573590279973

Use the examples entering the upper and lower limits of integration.