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Solving integrals/ (1/cos^2x-sinx)

Integral of (1/cos^2x-sinx) dx

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The solution

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 |  /   1            \   
 |  |------- - sin(x)| dx
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 |  \cos (x)         /   
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∫n4n4(−sin⁡(x)+1cos⁡2(x)) dx\int\limits_{\frac{n}{4}}^{\frac{n}{4}} \left(- \sin{\left(x \right)} + \frac{1}{\cos^{2}{\left(x \right)}}\right)\, dx 
Integral(1/(cos(x)^2) - sin(x), (x, n/4, n/4))
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:

      ∫(−sin⁡(x)) dx=−∫sin⁡(x) dx\int \left(- \sin{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)}\, dx

      1. The integral of sine is negative cosine:

        ∫sin⁡(x) dx=−cos⁡(x)\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}

      So, the result is: cos⁡(x)\cos{\left(x \right)}

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

      But the integral is

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

    The result is: sin⁡(x)cos⁡(x)+cos⁡(x)\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \cos{\left(x \right)}

  2. Now simplify:

    cos⁡(x)+tan⁡(x)\cos{\left(x \right)} + \tan{\left(x \right)}

  3. Add the constant of integration:

    cos⁡(x)+tan⁡(x)+constant\cos{\left(x \right)} + \tan{\left(x \right)}+ \mathrm{constant}

The answer is:

cos⁡(x)+tan⁡(x)+constant\cos{\left(x \right)} + \tan{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                           
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 | /   1            \          sin(x)         
 | |------- - sin(x)| dx = C + ------ + cos(x)
 | |   2            |          cos(x)         
 | \cos (x)         /                         
 |                                            
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∫(−sin⁡(x)+1cos⁡2(x)) dx=C+sin⁡(x)cos⁡(x)+cos⁡(x)\int \left(- \sin{\left(x \right)} + \frac{1}{\cos^{2}{\left(x \right)}}\right)\, dx = C + \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \cos{\left(x \right)}
Simplify
The answer [src] 
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Numerical answer [src] 
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    Use the examples entering the upper and lower limits of integration.

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