Mister Exam

Other calculators

  • How to use it?

  • Integral of d{x}:
  • Integral of e*x Integral of e*x
  • Integral of (1+4x^2)^(1/2) Integral of (1+4x^2)^(1/2)
  • Integral of x/(3x²-1)^4 Integral of x/(3x²-1)^4
  • Integral of (x+2)e^(-x) Integral of (x+2)e^(-x)

  • Identical expressions

  • log(atan(x))/(one +x^ two)
  • logarithm of ( arc tangent of gent of (x)) divide by (1 plus x squared )
  • logarithm of ( arc tangent of gent of (x)) divide by (one plus x to the power of two)
  • log(atan(x))/(1+x2)
  • logatanx/1+x2
  • log(atan(x))/(1+x²)
  • log(atan(x))/(1+x to the power of 2)
  • logatanx/1+x^2
  • log(atan(x)) divide by (1+x^2)
  • log(atan(x))/(1+x^2)dx

  • Similar expressions

  • log(atan(x))/(1-x^2)
  • log(arctan(x))/(1+x^2)
  • log(arctanx)/(1+x^2)
Solving integrals/ log(atan(x))/(1+x^2)

Integral of log(atan(x))/(1+x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                
  /                
 |                 
 |  log(atan(x))   
 |  ------------ dx
 |          2      
 |     1 + x       
 |                 
/                  
0
∫01log⁡(atan⁡(x))x2+1 dx\int\limits_{0}^{1} \frac{\log{\left(\operatorname{atan}{\left(x \right)} \right)}}{x^{2} + 1}\, dx 
Integral(log(atan(x))/(1 + x^2), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=log⁡(atan⁡(x))u = \log{\left(\operatorname{atan}{\left(x \right)} \right)}.

      Then let du=dx(x2+1)atan⁡(x)du = \frac{dx}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} and substitute dudu:

      ∫ueu du\int u e^{u}\, du

      1. Use integration by parts:

        ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

        Let u(u)=uu{\left(u \right)} = u and let dv⁡(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

        Then du⁡(u)=1\operatorname{du}{\left(u \right)} = 1.

        To find v(u)v{\left(u \right)}:

        1. The integral of the exponential function is itself.

          ∫eu du=eu\int e^{u}\, du = e^{u}

        Now evaluate the sub-integral.

      2. The integral of the exponential function is itself.

        ∫eu du=eu\int e^{u}\, du = e^{u}

      Now substitute uu back in:

      log⁡(atan⁡(x))atan⁡(x)−atan⁡(x)\log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} - \operatorname{atan}{\left(x \right)}

    Method #2

    1. Use integration by parts:

      ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(x)=log⁡(atan⁡(x))u{\left(x \right)} = \log{\left(\operatorname{atan}{\left(x \right)} \right)} and let dv⁡(x)=1x2+1\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} + 1}.

      Then du⁡(x)=1(x2+1)atan⁡(x)\operatorname{du}{\left(x \right)} = \frac{1}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}.

      To find v(x)v{\left(x \right)}:

      1. The integral of 1x2+1\frac{1}{x^{2} + 1} is atan⁡(x)\operatorname{atan}{\left(x \right)}.

      Now evaluate the sub-integral.

    2. The integral of 1x2+1\frac{1}{x^{2} + 1} is atan⁡(x)\operatorname{atan}{\left(x \right)}.

  2. Now simplify:

    (log⁡(atan⁡(x))−1)atan⁡(x)\left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} - 1\right) \operatorname{atan}{\left(x \right)}

  3. Add the constant of integration:

    (log⁡(atan⁡(x))−1)atan⁡(x)+constant\left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} - 1\right) \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer is:

(log⁡(atan⁡(x))−1)atan⁡(x)+constant\left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} - 1\right) \operatorname{atan}{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                    
 |                                                     
 | log(atan(x))                                        
 | ------------ dx = C - atan(x) + atan(x)*log(atan(x))
 |         2                                           
 |    1 + x                                            
 |                                                     
/
arctan⁡x log⁡arctan⁡x−arctan⁡x\arctan x\,\log \arctan x-\arctan x
Simplify
The answer [src] 
             /pi\
       pi*log|--|
  pi         \4 /
- -- + ----------
  4        4
π log⁡(π4)4−π4{{\pi\,\log \left({{\pi}\over{4}}\right)}\over{4}}-{{\pi}\over{4}} 
=
= 
             /pi\
       pi*log|--|
  pi         \4 /
- -- + ----------
  4        4
−π4+πlog⁡(π4)4- \frac{\pi}{4} + \frac{\pi \log{\left(\frac{\pi}{4} \right)}}{4}
Simplify
Numerical answer [src] 
-0.97512245861696
-0.97512245861696

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

    © Mister Exam – Calculator