Mister Exam

Other calculators

cos³xsin²x
Solving integrals/ cos³xsin²x

Integral of cos³xsin²x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                   
  /                   
 |                    
 |     3       2      
 |  cos (x)*sin (x) dx
 |                    
/                     
0
∫01sin⁡2(x)cos⁡3(x) dx\int\limits_{0}^{1} \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx 
Integral(cos(x)^3*sin(x)^2, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

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

      ∫(−u4+u2) du\int \left(- u^{4} + u^{2}\right)\, du

      1. Integrate term-by-term:

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

          ∫(−u4) du=−∫u4 du\int \left(- u^{4}\right)\, du = - \int u^{4}\, du

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

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          So, the result is: −u55- \frac{u^{5}}{5}

        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}

        The result is: −u55+u33- \frac{u^{5}}{5} + \frac{u^{3}}{3}

      Now substitute uu back in:

      −sin⁡5(x)5+sin⁡3(x)3- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}

    Method #2

    1. Rewrite the integrand:

      (1−sin⁡2(x))sin⁡2(x)cos⁡(x)=−sin⁡4(x)cos⁡(x)+sin⁡2(x)cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} = - \sin^{4}{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\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⁡4(x)cos⁡(x)) dx=−∫sin⁡4(x)cos⁡(x) dx\int \left(- \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{4}{\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:

          ∫u4 du\int u^{4}\, du

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

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          Now substitute uu back in:

          sin⁡5(x)5\frac{\sin^{5}{\left(x \right)}}{5}

        So, the result is: −sin⁡5(x)5- \frac{\sin^{5}{\left(x \right)}}{5}

      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}

      The result is: −sin⁡5(x)5+sin⁡3(x)3- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}

    Method #3

    1. Rewrite the integrand:

      (1−sin⁡2(x))sin⁡2(x)cos⁡(x)=−sin⁡4(x)cos⁡(x)+sin⁡2(x)cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)} = - \sin^{4}{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\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⁡4(x)cos⁡(x)) dx=−∫sin⁡4(x)cos⁡(x) dx\int \left(- \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{4}{\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:

          ∫u4 du\int u^{4}\, du

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

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          Now substitute uu back in:

          sin⁡5(x)5\frac{\sin^{5}{\left(x \right)}}{5}

        So, the result is: −sin⁡5(x)5- \frac{\sin^{5}{\left(x \right)}}{5}

      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}

      The result is: −sin⁡5(x)5+sin⁡3(x)3- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}

  3. Add the constant of integration:

    −sin⁡5(x)5+sin⁡3(x)3+constant- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}+ \mathrm{constant}

The answer is:

−sin⁡5(x)5+sin⁡3(x)3+constant- \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                          
 |                             5         3   
 |    3       2             sin (x)   sin (x)
 | cos (x)*sin (x) dx = C - ------- + -------
 |                             5         3   
/
−3 sin⁡5x−5 sin⁡3x15-{{3\,\sin ^5x-5\,\sin ^3x}\over{15}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.00.2
Plot the graph f(x)
The answer [src] 
     5         3   
  sin (1)   sin (1)
- ------- + -------
     5         3
−3 sin⁡51−5 sin⁡3115-{{3\,\sin ^51-5\,\sin ^31}\over{15}} 
=
= 
     5         3   
  sin (1)   sin (1)
- ------- + -------
     5         3
−sin⁡5(1)5+sin⁡3(1)3- \frac{\sin^{5}{\left(1 \right)}}{5} + \frac{\sin^{3}{\left(1 \right)}}{3}
Simplify
Numerical answer [src] 
0.114230426366362
0.114230426366362
The graph
Integral of cos³xsin²x dx

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

    © Mister Exam – Calculator