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sinx+sin^2x+sin^3x

Integral of sinx+sin^2x+sin^3x dx

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                                
  /                                
 |                                 
 |  /            2         3   \   
 |  \sin(x) + sin (x) + sin (x)/ dx
 |                                 
/                                  
0                                  
$$\int\limits_{0}^{1} \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) + \sin^{3}{\left(x \right)}\right)\, dx$$
Integral(sin(x) + sin(x)^2 + sin(x)^3, (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

    1. Integrate term-by-term:

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

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

          1. Let .

            Then let and substitute :

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

              1. The integral of cosine is sine:

              So, the result is:

            Now substitute back in:

          So, the result is:

        The result is:

      1. The integral of sine is negative cosine:

      The result is:

    1. Rewrite the integrand:

    2. There are multiple ways to do this integral.

      Method #1

      1. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          1. The integral of is when :

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

          The result is:

        Now substitute back in:

      Method #2

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

          1. Let .

            Then let and substitute :

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

              1. The integral of is when :

              So, the result is:

            Now substitute back in:

          So, the result is:

        1. The integral of sine is negative cosine:

        The result is:

      Method #3

      1. Rewrite the integrand:

      2. Integrate term-by-term:

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

          1. Let .

            Then let and substitute :

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

              1. The integral of is when :

              So, the result is:

            Now substitute back in:

          So, the result is:

        1. The integral of sine is negative cosine:

        The result is:

    The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                       
 |                                                                    3   
 | /            2         3   \          x              sin(2*x)   cos (x)
 | \sin(x) + sin (x) + sin (x)/ dx = C + - - 2*cos(x) - -------- + -------
 |                                       2                 4          3   
/                                                                         
$$\int \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)}\right) + \sin^{3}{\left(x \right)}\right)\, dx = C + \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} + \frac{\cos^{3}{\left(x \right)}}{3} - 2 \cos{\left(x \right)}$$
The graph
The answer [src]
                   3                   
13              cos (1)   cos(1)*sin(1)
-- - 2*cos(1) + ------- - -------------
6                  3            2      
$$- 2 \cos{\left(1 \right)} - \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{3}{\left(1 \right)}}{3} + \frac{13}{6}$$
=
=
                   3                   
13              cos (1)   cos(1)*sin(1)
-- - 2*cos(1) + ------- - -------------
6                  3            2      
$$- 2 \cos{\left(1 \right)} - \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{3}{\left(1 \right)}}{3} + \frac{13}{6}$$
13/6 - 2*cos(1) + cos(1)^3/3 - cos(1)*sin(1)/2
Numerical answer [src]
0.911313899974298
0.911313899974298
The graph
Integral of sinx+sin^2x+sin^3x dx

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