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Solving integrals/ (1/sin²x-4x)

Integral of (1/sin²x-4x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  0                   
  /                   
 |                    
 |  /   1         \   
 |  |------- - 4*x| dx
 |  |   2         |   
 |  \sin (x)      /   
 |                    
/                     
pi
∫π0(−4x+1sin⁡2(x)) dx\int\limits_{\pi}^{0} \left(- 4 x + \frac{1}{\sin^{2}{\left(x \right)}}\right)\, dx 
Integral(1/(sin(x)^2) - 4*x, (x, pi, 0))
Detail solution

  1. Integrate term-by-term:

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

      ∫(−4x) dx=−4∫x dx\int \left(- 4 x\right)\, dx = - 4 \int x\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      So, the result is: −2x2- 2 x^{2}

    1. Don't know the steps in finding this integral.

      But the integral is

      −cos⁡(x)sin⁡(x)- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

    The result is: −2x2−cos⁡(x)sin⁡(x)- 2 x^{2} - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

  2. Now simplify:

    −2x2−1tan⁡(x)- 2 x^{2} - \frac{1}{\tan{\left(x \right)}}

  3. Add the constant of integration:

    −2x2−1tan⁡(x)+constant- 2 x^{2} - \frac{1}{\tan{\left(x \right)}}+ \mathrm{constant}

The answer is:

−2x2−1tan⁡(x)+constant- 2 x^{2} - \frac{1}{\tan{\left(x \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                      
 |                                       
 | /   1         \             2   cos(x)
 | |------- - 4*x| dx = C - 2*x  - ------
 | |   2         |                 sin(x)
 | \sin (x)      /                       
 |                                       
/
∫(−4x+1sin⁡2(x)) dx=C−2x2−cos⁡(x)sin⁡(x)\int \left(- 4 x + \frac{1}{\sin^{2}{\left(x \right)}}\right)\, dx = C - 2 x^{2} - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}
Simplify
The graph
0.000.250.500.751.001.251.501.752.002.252.502.753.00-5e3110e31
Plot the graph f(x)
The answer [src] 
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−∞-\infty 
=
= 
-oo
−∞-\infty 
-oo
Numerical answer [src] 
-4.39812420049212e+18
-4.39812420049212e+18

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

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