Mister Exam

Integral of sinx*cos3x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                   
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 |  sin(x)*cos(3*x) dx
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$$\int\limits_{0}^{1} \sin{\left(x \right)} \cos{\left(3 x \right)}\, dx$$
Integral(sin(x)*cos(3*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. There are multiple ways to do this integral.

        Method #1

        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:

        Method #2

        1. Rewrite the integrand:

        2. Let .

          Then let and substitute :

          1. 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. 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. 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]
  /                                        2   
 |                             4      3*cos (x)
 | sin(x)*cos(3*x) dx = C - cos (x) + ---------
 |                                        2    
/                                              
$${{\cos \left(2\,x\right)}\over{4}}-{{\cos \left(4\,x\right)}\over{8 }}$$
The graph
The answer [src]
  1   cos(1)*cos(3)   3*sin(1)*sin(3)
- - + ------------- + ---------------
  8         8                8       
$$-{{\cos 4-2\,\cos 2}\over{8}}-{{1}\over{8}}$$
=
=
  1   cos(1)*cos(3)   3*sin(1)*sin(3)
- - + ------------- + ---------------
  8         8                8       
$$- \frac{1}{8} + \frac{\cos{\left(1 \right)} \cos{\left(3 \right)}}{8} + \frac{3 \sin{\left(1 \right)} \sin{\left(3 \right)}}{8}$$
Numerical answer [src]
-0.147331256528834
-0.147331256528834
The graph
Integral of sinx*cos3x dx

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