Integral of xcsc²x dx

The solution

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 |  x*csc (x) dx
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$$\int\limits_{0}^{1} x \csc^{2}{\left(x \right)}\, dx$$

Integral(x*csc(x)^2, (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \csc^{2}{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. $\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \cot{\left(x \right)}\right)\, dx = - \int \cot{\left(x \right)}\, dx$
1. Rewrite the integrand:
  
  $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
2. 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:
  
  $\log{\left(\sin{\left(x \right)} \right)}$
So, the result is: $- \log{\left(\sin{\left(x \right)} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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 | x*csc (x) dx = C - x*cot(x) + log(sin(x))
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$${{\left(\sin ^2\left(2\,x\right)+\cos ^2\left(2\,x\right)-2\,\cos \left(2\,x\right)+1\right)\,\log \left(\sin ^2x+\cos ^2x+2\,\cos x+1 \right)+\left(\sin ^2\left(2\,x\right)+\cos ^2\left(2\,x\right)-2\, \cos \left(2\,x\right)+1\right)\,\log \left(\sin ^2x+\cos ^2x-2\, \cos x+1\right)-4\,x\,\sin \left(2\,x\right)}\over{2\,\sin ^2\left(2 \,x\right)+2\,\cos ^2\left(2\,x\right)-4\,\cos \left(2\,x\right)+2}}$$

Simplify

The answer [src]

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$${\it \%a}$$

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$$\infty$$

Simplify

Numerical answer [src]

44.2757497717895

44.2757497717895

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

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Identical expressions

Integral of xcsc²x dx

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The solution