Mister Exam

Integral of sin5xcosxdx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                     
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 |  sin(5*x)*cos(x)*1 dx
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$$\int\limits_{0}^{1} \sin{\left(5 x \right)} \cos{\left(x \right)} 1\, dx$$
Integral(sin(5*x)*cos(x)*1, (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. There are multiple ways to do this integral.

        Method #1

        1. Let .

          Then let and substitute :

          1. The integral of is when :

          Now substitute back in:

        Method #2

        1. Rewrite the integrand:

        2. 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 is when :

        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:

    The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                            2           6   
 |                                 4      5*cos (x)   8*sin (x)
 | sin(5*x)*cos(x)*1 dx = C - 5*sin (x) - --------- + ---------
 |                                            2           3    
/                                                              
$$\int \sin{\left(5 x \right)} \cos{\left(x \right)} 1\, dx = C + \frac{8 \sin^{6}{\left(x \right)}}{3} - 5 \sin^{4}{\left(x \right)} - \frac{5 \cos^{2}{\left(x \right)}}{2}$$
The graph
The answer [src]
5    5*cos(1)*cos(5)   sin(1)*sin(5)
-- - --------------- - -------------
24          24               24     
$$- \frac{5 \cos{\left(1 \right)} \cos{\left(5 \right)}}{24} - \frac{\sin{\left(1 \right)} \sin{\left(5 \right)}}{24} + \frac{5}{24}$$
=
=
5    5*cos(1)*cos(5)   sin(1)*sin(5)
-- - --------------- - -------------
24          24               24     
$$- \frac{5 \cos{\left(1 \right)} \cos{\left(5 \right)}}{24} - \frac{\sin{\left(1 \right)} \sin{\left(5 \right)}}{24} + \frac{5}{24}$$
Numerical answer [src]
0.210024595387088
0.210024595387088
The graph
Integral of sin5xcosxdx dx

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