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(sin^5x+cos^4x)

Integral of (sin^5x+cos^4x) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  pi                       
  --                       
  4                        
   /                       
  |                        
  |  /   5         4   \   
  |  \sin (x) + cos (x)/ dx
  |                        
 /                         
-pi                        
----                       
 4                         
$$\int\limits_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \left(\sin^{5}{\left(x \right)} + \cos^{4}{\left(x \right)}\right)\, dx$$
Integral(sin(x)^5 + cos(x)^4, (x, -pi/4, pi/4))
Detail solution
  1. Integrate term-by-term:

    1. Rewrite the integrand:

    2. There are multiple ways to do this integral.

      Method #1

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        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:

        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 #2

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        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:

        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:

    1. Rewrite the integrand:

    2. There are multiple ways to do this integral.

      Method #1

      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. 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 cosine is sine:

                  So, the result is:

                Now substitute back in:

              So, the result is:

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

            The result is:

          So, the result is:

        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:

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

        The result is:

      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. 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 cosine is sine:

                  So, the result is:

                Now substitute back in:

              So, the result is:

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

            The result is:

          So, the result is:

        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:

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

        The result is:

    The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                     
 |                                          5                                 3         
 | /   5         4   \                   cos (x)   sin(2*x)   sin(4*x)   2*cos (x)   3*x
 | \sin (x) + cos (x)/ dx = C - cos(x) - ------- + -------- + -------- + --------- + ---
 |                                          5         4          32          3        8 
/                                                                                       
$${{{{{{\sin \left(4\,x\right)}\over{2}}+2\,x}\over{8}}+{{\sin \left( 2\,x\right)}\over{2}}+{{x}\over{2}}}\over{2}}-{{\cos ^5x}\over{5}}+ {{2\,\cos ^3x}\over{3}}-\cos x$$
The graph
The answer [src]
1   3*pi
- + ----
2    16 
$${{\sin \pi+3\,\pi+8\,\sin \left({{\pi}\over{2}}\right)}\over{16}}$$
=
=
1   3*pi
- + ----
2    16 
$$\frac{1}{2} + \frac{3 \pi}{16}$$
Numerical answer [src]
1.08904862254809
1.08904862254809
The graph
Integral of (sin^5x+cos^4x) dx

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