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cosx^3
Solving integrals/ cosx^3

Integral of cosx^3 dx

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∫01cos⁡3(x) dx\int\limits_{0}^{1} \cos^{3}{\left(x \right)}\, dx 
Integral(cos(x)^3, (x, 0, 1))
Detail solution

  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)}

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                 
 |                     3            
 |    3             sin (x)         
 | cos (x) dx = C - ------- + sin(x)
 |                     3            
/
∫cos⁡3(x) dx=C−sin⁡3(x)3+sin⁡(x)\int \cos^{3}{\left(x \right)}\, dx = C - \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
     3            
  sin (1)         
- ------- + sin(1)
     3
−sin⁡3(1)3+sin⁡(1)- \frac{\sin^{3}{\left(1 \right)}}{3} + \sin{\left(1 \right)} 
=
= 
     3            
  sin (1)         
- ------- + sin(1)
     3
−sin⁡3(1)3+sin⁡(1)- \frac{\sin^{3}{\left(1 \right)}}{3} + \sin{\left(1 \right)} 
-sin(1)^3/3 + sin(1)
Numerical answer [src] 
0.642863239277578
0.642863239277578
The graph
Integral of cosx^3 dx

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

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