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Solving integrals/ cot(sqrt(x)+1)*(1/sqrt(x))

Integral of cot(sqrt(x)+1)*(1/sqrt(x)) dx

Limits of integration:

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The solution

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∫01cot⁡(x+1)x dx\int\limits_{0}^{1} \frac{\cot{\left(\sqrt{x} + 1 \right)}}{\sqrt{x}}\, dx 
Integral(cot(sqrt(x) + 1)/sqrt(x), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x+1u = \sqrt{x} + 1.

      Then let du=dx2xdu = \frac{dx}{2 \sqrt{x}} and substitute 2du2 du:

      ∫2cot⁡(u) du\int 2 \cot{\left(u \right)}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫cot⁡(u) du=2∫cot⁡(u) du\int \cot{\left(u \right)}\, du = 2 \int \cot{\left(u \right)}\, du

        1. Rewrite the integrand:

          cot⁡(u)=cos⁡(u)sin⁡(u)\cot{\left(u \right)} = \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}

        2. Let u=sin⁡(u)u = \sin{\left(u \right)}.

          Then let du=cos⁡(u)dudu = \cos{\left(u \right)} du and substitute dudu:

          ∫1u du\int \frac{1}{u}\, du

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          Now substitute uu back in:

          log⁡(sin⁡(u))\log{\left(\sin{\left(u \right)} \right)}

        So, the result is: 2log⁡(sin⁡(u))2 \log{\left(\sin{\left(u \right)} \right)}

      Now substitute uu back in:

      2log⁡(sin⁡(x+1))2 \log{\left(\sin{\left(\sqrt{x} + 1 \right)} \right)}

    Method #2

    1. Let u=xu = \sqrt{x}.

      Then let du=dx2xdu = \frac{dx}{2 \sqrt{x}} and substitute 2du2 du:

      ∫2cot⁡(u+1) du\int 2 \cot{\left(u + 1 \right)}\, du

      1. The integral of a constant times a function is the constant times the integral of the function:

        ∫cot⁡(u+1) du=2∫cot⁡(u+1) du\int \cot{\left(u + 1 \right)}\, du = 2 \int \cot{\left(u + 1 \right)}\, du

        1. Rewrite the integrand:

          cot⁡(u+1)=cos⁡(u+1)sin⁡(u+1)\cot{\left(u + 1 \right)} = \frac{\cos{\left(u + 1 \right)}}{\sin{\left(u + 1 \right)}}

        2. Let u=sin⁡(u+1)u = \sin{\left(u + 1 \right)}.

          Then let du=cos⁡(u+1)dudu = \cos{\left(u + 1 \right)} du and substitute dudu:

          ∫1u du\int \frac{1}{u}\, du

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          Now substitute uu back in:

          log⁡(sin⁡(u+1))\log{\left(\sin{\left(u + 1 \right)} \right)}

        So, the result is: 2log⁡(sin⁡(u+1))2 \log{\left(\sin{\left(u + 1 \right)} \right)}

      Now substitute uu back in:

      2log⁡(sin⁡(x+1))2 \log{\left(\sin{\left(\sqrt{x} + 1 \right)} \right)}

  2. Now simplify:

    2log⁡(sin⁡(x+1))2 \log{\left(\sin{\left(\sqrt{x} + 1 \right)} \right)}

  3. Add the constant of integration:

    2log⁡(sin⁡(x+1))+constant2 \log{\left(\sin{\left(\sqrt{x} + 1 \right)} \right)}+ \mathrm{constant}

The answer is:

2log⁡(sin⁡(x+1))+constant2 \log{\left(\sin{\left(\sqrt{x} + 1 \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                             
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 |    /  ___    \                               
 | cot\\/ x  + 1/               /   /  ___    \\
 | -------------- dx = C + 2*log\sin\\/ x  + 1//
 |       ___                                    
 |     \/ x                                     
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∫cot⁡(x+1)x dx=C+2log⁡(sin⁡(x+1))\int \frac{\cot{\left(\sqrt{x} + 1 \right)}}{\sqrt{x}}\, dx = C + 2 \log{\left(\sin{\left(\sqrt{x} + 1 \right)} \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-50100
Plot the graph f(x)
Numerical answer [src] 
0.155041420007179
0.155041420007179

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

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