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Solving integrals/ 3cos^3x

Integral of 3cos^3x dx

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The solution

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∫013cos⁡3(x) dx\int\limits_{0}^{1} 3 \cos^{3}{\left(x \right)}\, dx 
Integral(3*cos(x)^3, (x, 0, 1))
Detail solution

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

    ∫3cos⁡3(x) dx=3∫cos⁡3(x) dx\int 3 \cos^{3}{\left(x \right)}\, dx = 3 \int \cos^{3}{\left(x \right)}\, dx

    1. Rewrite the integrand:

      cos⁡3(x)=(1−sin⁡2(x))cos⁡(x)\cos^{3}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}

    2. There are multiple ways to do this integral.

      Method #1

      1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

        Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

        ∫(1−u2) du\int \left(1 - u^{2}\right)\, du

        1. Integrate term-by-term:

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

            ∫1 du=u\int 1\, du = u

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

            ∫(−u2) du=−∫u2 du\int \left(- u^{2}\right)\, du = - \int u^{2}\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

            So, the result is: −u33- \frac{u^{3}}{3}

          The result is: −u33+u- \frac{u^{3}}{3} + u

        Now substitute uu back in:

        −sin⁡3(x)3+sin⁡(x)- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}

      Method #2

      1. Rewrite the integrand:

        (1−sin⁡2(x))cos⁡(x)=−sin⁡2(x)cos⁡(x)+cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}

      2. Integrate term-by-term:

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

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

          1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

            Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

            ∫u2 du\int u^{2}\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

            Now substitute uu back in:

            sin⁡3(x)3\frac{\sin^{3}{\left(x \right)}}{3}

          So, the result is: −sin⁡3(x)3- \frac{\sin^{3}{\left(x \right)}}{3}

        1. The integral of cosine is sine:

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

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

      Method #3

      1. Rewrite the integrand:

        (1−sin⁡2(x))cos⁡(x)=−sin⁡2(x)cos⁡(x)+cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}

      2. Integrate term-by-term:

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

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

          1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

            Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

            ∫u2 du\int u^{2}\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

            Now substitute uu back in:

            sin⁡3(x)3\frac{\sin^{3}{\left(x \right)}}{3}

          So, the result is: −sin⁡3(x)3- \frac{\sin^{3}{\left(x \right)}}{3}

        1. The integral of cosine is sine:

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

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

    So, the result is: −sin⁡3(x)+3sin⁡(x)- \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}

  2. Now simplify:

    (cos⁡2(x)+2)sin⁡(x)\left(\cos^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)}

  3. Add the constant of integration:

    (cos⁡2(x)+2)sin⁡(x)+constant\left(\cos^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)}+ \mathrm{constant}

The answer is:

(cos⁡2(x)+2)sin⁡(x)+constant\left(\cos^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                     
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 |      3                3              
 | 3*cos (x) dx = C - sin (x) + 3*sin(x)
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∫3cos⁡3(x) dx=C−sin⁡3(x)+3sin⁡(x)\int 3 \cos^{3}{\left(x \right)}\, dx = C - \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9005
Plot the graph f(x)
The answer [src] 
     3              
- sin (1) + 3*sin(1)
−sin⁡3(1)+3sin⁡(1)- \sin^{3}{\left(1 \right)} + 3 \sin{\left(1 \right)} 
=
= 
     3              
- sin (1) + 3*sin(1)
−sin⁡3(1)+3sin⁡(1)- \sin^{3}{\left(1 \right)} + 3 \sin{\left(1 \right)} 
-sin(1)^3 + 3*sin(1)
Numerical answer [src] 
1.92858971783273
1.92858971783273

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

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