Mister Exam

Integral of sin4xcosx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                   
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 |  sin(4*x)*cos(x) dx
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$$\int\limits_{0}^{1} \sin{\left(4 x \right)} \cos{\left(x \right)}\, dx$$
Integral(sin(4*x)*cos(x), (x, 0, 1))
Detail solution
  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. 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 times a function is the constant times the integral of the function:

              1. The integral of is when :

              So, the result is:

            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. 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:

          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. 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:

          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 is when :

          So, the result is:

        Now substitute back in:

      So, the result is:

    The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                              5           3   
 |                          8*cos (x)   4*cos (x)
 | sin(4*x)*cos(x) dx = C - --------- + ---------
 |                              5           3    
/                                                
$$-{{\cos \left(5\,x\right)}\over{10}}-{{\cos \left(3\,x\right) }\over{6}}$$
The graph
The answer [src]
4    4*cos(1)*cos(4)   sin(1)*sin(4)
-- - --------------- - -------------
15          15               15     
$${{4}\over{15}}-{{3\,\cos 5+5\,\cos 3}\over{30}}$$
=
=
4    4*cos(1)*cos(4)   sin(1)*sin(4)
-- - --------------- - -------------
15          15               15     
$$- \frac{\sin{\left(1 \right)} \sin{\left(4 \right)}}{15} - \frac{4 \cos{\left(1 \right)} \cos{\left(4 \right)}}{15} + \frac{4}{15}$$
Numerical answer [src]
0.403299197553752
0.403299197553752
The graph
Integral of sin4xcosx dx

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