Mister Exam

Other calculators


sin(7*x)*cos(4*x)

Integral of sin(7*x)*cos(4*x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                     
  /                     
 |                      
 |  sin(7*x)*cos(4*x) dx
 |                      
/                       
0                       
$$\int\limits_{0}^{1} \sin{\left(7 x \right)} \cos{\left(4 x \right)}\, dx$$
Integral(sin(7*x)*cos(4*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:

            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:

            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. 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 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. 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 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. Rewrite the integrand:

      2. 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:

          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:

          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:

      So, the result is:

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

      1. Rewrite the integrand:

      2. Rewrite the integrand:

      3. 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 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 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:

      So, the result is:

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

      1. Rewrite the integrand:

      2. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          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:

          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:

          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:

      So, the result is:

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

      1. Rewrite the integrand:

      2. Let .

        Then let and substitute :

        1. Integrate term-by-term:

          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:

          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:

          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:

      So, the result is:

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

      1. Rewrite the integrand:

      2. Rewrite the integrand:

      3. 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:

      So, the result is:

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

      1. Rewrite the integrand:

      2. 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:

      So, the result is:

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

      1. Rewrite the integrand:

      2. 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:

      So, the result is:

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

      1. Rewrite the integrand:

      2. 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:

      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:

    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 a constant times a function is the constant times the integral of the function:

      1. The integral of sine is negative cosine:

      So, the result is:

    The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                           11            3            
 |                                   7            5             9      512*cos  (x)   32*cos (x)         
 | sin(7*x)*cos(4*x) dx = C - 128*cos (x) + 56*cos (x) + 128*cos (x) - ------------ - ---------- + cos(x)
 |                                                                          11            3              
/                                                                                                        
$$\int \sin{\left(7 x \right)} \cos{\left(4 x \right)}\, dx = C - \frac{512 \cos^{11}{\left(x \right)}}{11} + 128 \cos^{9}{\left(x \right)} - 128 \cos^{7}{\left(x \right)} + 56 \cos^{5}{\left(x \right)} - \frac{32 \cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}$$
The graph
The answer [src]
7    7*cos(4)*cos(7)   4*sin(4)*sin(7)
-- - --------------- - ---------------
33          33                33      
$$- \frac{4 \sin{\left(4 \right)} \sin{\left(7 \right)}}{33} - \frac{7 \cos{\left(4 \right)} \cos{\left(7 \right)}}{33} + \frac{7}{33}$$
=
=
7    7*cos(4)*cos(7)   4*sin(4)*sin(7)
-- - --------------- - ---------------
33          33                33      
$$- \frac{4 \sin{\left(4 \right)} \sin{\left(7 \right)}}{33} - \frac{7 \cos{\left(4 \right)} \cos{\left(7 \right)}}{33} + \frac{7}{33}$$
Numerical answer [src]
0.376918793464254
0.376918793464254
The graph
Integral of sin(7*x)*cos(4*x) dx

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