Mister Exam

Integral of 15xarctgxdx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                
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 |  15*x*atan(x) dx
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$$\int\limits_{0}^{1} 15 x \operatorname{atan}{\left(x \right)}\, dx$$
Integral((15*x)*atan(x), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

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

      1. The integral of is when :

      So, the result is:

    Now evaluate the sub-integral.

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

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of a constant is the constant times the variable of integration:

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

          PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)

        So, the result is:

      The result is:

    So, the result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                              2        
 |                       15*x   15*atan(x)   15*x *atan(x)
 | 15*x*atan(x) dx = C - ---- + ---------- + -------------
 |                        2         2              2      
/                                                         
$$\int 15 x \operatorname{atan}{\left(x \right)}\, dx = C + \frac{15 x^{2} \operatorname{atan}{\left(x \right)}}{2} - \frac{15 x}{2} + \frac{15 \operatorname{atan}{\left(x \right)}}{2}$$
The graph
The answer [src]
  15   15*pi
- -- + -----
  2      4  
$$- \frac{15}{2} + \frac{15 \pi}{4}$$
=
=
  15   15*pi
- -- + -----
  2      4  
$$- \frac{15}{2} + \frac{15 \pi}{4}$$
-15/2 + 15*pi/4
Numerical answer [src]
4.28097245096173
4.28097245096173

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