Mister Exam

Other calculators


(-2x^2-12x+6)cos6x

Integral of (-2x^2-12x+6)cos6x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                                
  /                                
 |                                 
 |  /     2           \            
 |  \- 2*x  - 12*x + 6/*cos(6*x) dx
 |                                 
/                                  
0                                  
$$\int\limits_{0}^{1} \left(- 2 x^{2} - 12 x + 6\right) \cos{\left(6 x \right)}\, dx$$
Integral((-2*x^2 - 12*x + 6)*cos(6*x), (x, 0, 1))
Detail solution
  1. 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. Use integration by parts:

          Let and let .

          Then .

          To find :

          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:

          Now evaluate the sub-integral.

        2. Use integration by parts:

          Let and let .

          Then .

          To find :

          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 sine is negative cosine:

              So, the result is:

            Now substitute back in:

          Now evaluate the sub-integral.

        3. 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:

        So, the result is:

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

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          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:

          Now evaluate the sub-integral.

        2. 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 sine is negative cosine:

              So, the result is:

            Now substitute back in:

          So, 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:

      The result is:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      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:

      Now evaluate the sub-integral.

    2. Use integration by parts:

      Let and let .

      Then .

      To find :

      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 sine is negative cosine:

          So, the result is:

        Now substitute back in:

      Now evaluate the sub-integral.

    3. 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:

    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. Use integration by parts:

          Let and let .

          Then .

          To find :

          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:

          Now evaluate the sub-integral.

        2. Use integration by parts:

          Let and let .

          Then .

          To find :

          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 sine is negative cosine:

              So, the result is:

            Now substitute back in:

          Now evaluate the sub-integral.

        3. 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:

        So, the result is:

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

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          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:

          Now evaluate the sub-integral.

        2. 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 sine is negative cosine:

              So, the result is:

            Now substitute back in:

          So, 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:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                                      
 |                                                                                2                      
 | /     2           \                   cos(6*x)   55*sin(6*x)                  x *sin(6*x)   x*cos(6*x)
 | \- 2*x  - 12*x + 6/*cos(6*x) dx = C - -------- + ----------- - 2*x*sin(6*x) - ----------- - ----------
 |                                          3            54                           3            9     
/                                                                                                        
$${{-{{\left(36\,x^2-2\right)\,\sin \left(6\,x\right)+12\,x\,\cos \left(6\,x\right)}\over{18}}-2\,\left(6\,x\,\sin \left(6\,x\right)+ \cos \left(6\,x\right)\right)+6\,\sin \left(6\,x\right)}\over{6}}$$
The graph
The answer [src]
1   71*sin(6)   4*cos(6)
- - --------- - --------
3       54         9    
$${{1}\over{3}}-{{71\,\sin 6+24\,\cos 6}\over{54}}$$
=
=
1   71*sin(6)   4*cos(6)
- - --------- - --------
3       54         9    
$$- \frac{4 \cos{\left(6 \right)}}{9} + \frac{1}{3} - \frac{71 \sin{\left(6 \right)}}{54}$$
Numerical answer [src]
0.273970620231758
0.273970620231758
The graph
Integral of (-2x^2-12x+6)cos6x dx

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