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Integral of d{x}:
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Identical expressions
x^ two *ctg(x^ three)
x squared multiply by ctg(x cubed )
x to the power of two multiply by ctg(x to the power of three)
x2*ctg(x3)
x2*ctgx3
x²*ctg(x³)
x to the power of 2*ctg(x to the power of 3)
x^2ctg(x^3)
x2ctg(x3)
x2ctgx3
x^2ctgx^3
x^2*ctg(x^3)dx

Solving integrals/ x^2*ctg(x^3)

Integral of x^2*ctg(x^3) dx

The solution

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 |   2    / 3\   
 |  x *cot\x / dx
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0

$$\int\limits_{0}^{1} x^{2} \cot{\left(x^{3} \right)}\, dx$$

Integral(x^2*cot(x^3), (x, 0, 1))

Detail solution

Let $u = x^{3}$ .

Then let $du = 3 x^{2} dx$ and substitute $\frac{du}{3}$ :

$\int \frac{\cot{\left(u \right)}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \cot{\left(u \right)}\, du = \frac{\int \cot{\left(u \right)}\, du}{3}$
  1. Rewrite the integrand:
    
    $\cot{\left(u \right)} = \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}$
  2. Let $u = \sin{\left(u \right)}$ .
    
    Then let $du = \cos{\left(u \right)} du$ 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:
    
    $\log{\left(\sin{\left(u \right)} \right)}$
  So, the result is: $\frac{\log{\left(\sin{\left(u \right)} \right)}}{3}$
Now substitute $u$ back in:

$\frac{\log{\left(\sin{\left(x^{3} \right)} \right)}}{3}$
Add the constant of integration:

$\frac{\log{\left(\sin{\left(x^{3} \right)} \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\sin{\left(x^{3} \right)} \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                        /   / 3\\
 |  2    / 3\          log\sin\x //
 | x *cot\x / dx = C + ------------
 |                          3      
/

$$\int x^{2} \cot{\left(x^{3} \right)}\, dx = C + \frac{\log{\left(\sin{\left(x^{3} \right)} \right)}}{3}$$

Simplify

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Numerical answer [src]

44.0329115519032

44.0329115519032

The graph

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

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Identical expressions

Integral of x^2*ctg(x^3) dx

Limits of integration:

The graph:

Piecewise:

The solution