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(sin3x-1/cos^2x)
Solving integrals/ (sin3x-1/cos^2x)

Integral of (sin3x-1/cos^2x) dx

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∫00(sin⁡(3x)−1⋅1cos⁡2(x)) dx\int\limits_{0}^{0} \left(\sin{\left(3 x \right)} - 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}}\right)\, dx
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Detail solution

  1. Integrate term-by-term:

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      ∫sin⁡(u)9 du\int \frac{\sin{\left(u \right)}}{9}\, du

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

        ∫sin⁡(u)3 du=∫sin⁡(u) du3\int \frac{\sin{\left(u \right)}}{3}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}

        1. The integral of sine is negative cosine:

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

        So, the result is: −cos⁡(u)3- \frac{\cos{\left(u \right)}}{3}

      Now substitute uu back in:

      −cos⁡(3x)3- \frac{\cos{\left(3 x \right)}}{3}

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

      ∫(−1cos⁡2(x)) dx=−∫1cos⁡2(x) dx\int \left(- \frac{1}{\cos^{2}{\left(x \right)}}\right)\, dx = - \int \frac{1}{\cos^{2}{\left(x \right)}}\, dx

      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)}}

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

    The result is: −sin⁡(x)cos⁡(x)−cos⁡(3x)3- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\cos{\left(3 x \right)}}{3}

  2. Now simplify:

    −cos⁡(3x)3−tan⁡(x)- \frac{\cos{\left(3 x \right)}}{3} - \tan{\left(x \right)}

  3. Add the constant of integration:

    −cos⁡(3x)3−tan⁡(x)+constant- \frac{\cos{\left(3 x \right)}}{3} - \tan{\left(x \right)}+ \mathrm{constant}

The answer is:

−cos⁡(3x)3−tan⁡(x)+constant- \frac{\cos{\left(3 x \right)}}{3} - \tan{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                 
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 | /                1   \          cos(3*x)   sin(x)
 | |sin(3*x) - 1*-------| dx = C - -------- - ------
 | |                2   |             3       cos(x)
 | \             cos (x)/                           
 |                                                  
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∫(sin⁡(3x)−1⋅1cos⁡2(x)) dx=C−cos⁡(3x)3−sin⁡(x)cos⁡(x)\int \left(\sin{\left(3 x \right)} - 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}}\right)\, dx = C - \frac{\cos{\left(3 x \right)}}{3} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1.50.0
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Integral of (sin3x-1/cos^2x) dx

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

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