Mister Exam

Other calculators

sin^4xcos^3xdx
Solving integrals/ sin^4xcos^3xdx

Integral of sin^4xcos^3xdx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                     
  /                     
 |                      
 |     4       3        
 |  sin (x)*cos (x)*1 dx
 |                      
/                       
0
∫01sin⁡4(x)cos⁡3(x)1 dx\int\limits_{0}^{1} \sin^{4}{\left(x \right)} \cos^{3}{\left(x \right)} 1\, dx 
Integral(sin(x)^4*cos(x)^3*1, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    sin⁡4(x)cos⁡3(x)1=(1−sin⁡2(x))sin⁡4(x)cos⁡(x)\sin^{4}{\left(x \right)} \cos^{3}{\left(x \right)} 1 = \left(1 - \sin^{2}{\left(x \right)}\right) \sin^{4}{\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:

      ∫(−u6+u4) du\int \left(- u^{6} + u^{4}\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:

          ∫(−u6) du=−∫u6 du\int \left(- u^{6}\right)\, du = - \int u^{6}\, du

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

            ∫u6 du=u77\int u^{6}\, du = \frac{u^{7}}{7}

          So, the result is: −u77- \frac{u^{7}}{7}

        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}

        The result is: −u77+u55- \frac{u^{7}}{7} + \frac{u^{5}}{5}

      Now substitute uu back in:

      −sin⁡7(x)7+sin⁡5(x)5- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{\sin^{5}{\left(x \right)}}{5}

    Method #2

    1. Rewrite the integrand:

      (1−sin⁡2(x))sin⁡4(x)cos⁡(x)=−sin⁡6(x)cos⁡(x)+sin⁡4(x)cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{4}{\left(x \right)} \cos{\left(x \right)} = - \sin^{6}{\left(x \right)} \cos{\left(x \right)} + \sin^{4}{\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⁡6(x)cos⁡(x)) dx=−∫sin⁡6(x)cos⁡(x) dx\int \left(- \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{6}{\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:

          ∫u6 du\int u^{6}\, du

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

            ∫u6 du=u77\int u^{6}\, du = \frac{u^{7}}{7}

          Now substitute uu back in:

          sin⁡7(x)7\frac{\sin^{7}{\left(x \right)}}{7}

        So, the result is: −sin⁡7(x)7- \frac{\sin^{7}{\left(x \right)}}{7}

      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}

      The result is: −sin⁡7(x)7+sin⁡5(x)5- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{\sin^{5}{\left(x \right)}}{5}

    Method #3

    1. Rewrite the integrand:

      (1−sin⁡2(x))sin⁡4(x)cos⁡(x)=−sin⁡6(x)cos⁡(x)+sin⁡4(x)cos⁡(x)\left(1 - \sin^{2}{\left(x \right)}\right) \sin^{4}{\left(x \right)} \cos{\left(x \right)} = - \sin^{6}{\left(x \right)} \cos{\left(x \right)} + \sin^{4}{\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⁡6(x)cos⁡(x)) dx=−∫sin⁡6(x)cos⁡(x) dx\int \left(- \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{6}{\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:

          ∫u6 du\int u^{6}\, du

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

            ∫u6 du=u77\int u^{6}\, du = \frac{u^{7}}{7}

          Now substitute uu back in:

          sin⁡7(x)7\frac{\sin^{7}{\left(x \right)}}{7}

        So, the result is: −sin⁡7(x)7- \frac{\sin^{7}{\left(x \right)}}{7}

      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}

      The result is: −sin⁡7(x)7+sin⁡5(x)5- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{\sin^{5}{\left(x \right)}}{5}

  3. Add the constant of integration:

    −sin⁡7(x)7+sin⁡5(x)5+constant- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{\sin^{5}{\left(x \right)}}{5}+ \mathrm{constant}

The answer is:

−sin⁡7(x)7+sin⁡5(x)5+constant- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{\sin^{5}{\left(x \right)}}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                            
 |                               7         5   
 |    4       3               sin (x)   sin (x)
 | sin (x)*cos (x)*1 dx = C - ------- + -------
 |                               7         5   
/
−5 sin⁡7x−7 sin⁡5x35-{{5\,\sin ^7x-7\,\sin ^5x}\over{35}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.000.10
Plot the graph f(x)
The answer [src] 
     7         5   
  sin (1)   sin (1)
- ------- + -------
     7         5
−5 sin⁡71−7 sin⁡5135-{{5\,\sin ^71-7\,\sin ^51}\over{35}} 
=
= 
     7         5   
  sin (1)   sin (1)
- ------- + -------
     7         5
−sin⁡7(1)7+sin⁡5(1)5- \frac{\sin^{7}{\left(1 \right)}}{7} + \frac{\sin^{5}{\left(1 \right)}}{5}
Simplify
Numerical answer [src] 
0.0417020785888258
0.0417020785888258
The graph
Integral of sin^4xcos^3xdx dx

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

    © Mister Exam – Calculator