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x^6sinx
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x^6sinx
x to the power of 6 sinus of x
x6sinx
x⁶sinx
x^6sinxdx

Integral of x^6sinx dx

The solution

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  pi              
  --              
  4               
   /              
  |               
  |    6          
  |   x *sin(x) dx
  |               
 /                
-2020

$$\int\limits_{-2020}^{\frac{\pi}{4}} x^{6} \sin{\left(x \right)}\, dx$$

Integral(x^6*sin(x), (x, -2020, pi/4))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x^{6}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 6 x^{5}$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = - 6 x^{5}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - 30 x^{4}$ .

To find $v{\left(x \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = - 30 x^{4}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - 120 x^{3}$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = 120 x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 360 x^{2}$ .

To find $v{\left(x \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = 360 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 720 x$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- 720 x \cos{\left(x \right)}\right)\, dx = - 720 \int x \cos{\left(x \right)}\, dx$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  Now evaluate the sub-integral.
2. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
So, the result is: $- 720 x \sin{\left(x \right)} - 720 \cos{\left(x \right)}$
Add the constant of integration:

$- x^{6} \cos{\left(x \right)} + 6 x^{5} \sin{\left(x \right)} + 30 x^{4} \cos{\left(x \right)} - 120 x^{3} \sin{\left(x \right)} - 360 x^{2} \cos{\left(x \right)} + 720 x \sin{\left(x \right)} + 720 \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- x^{6} \cos{\left(x \right)} + 6 x^{5} \sin{\left(x \right)} + 30 x^{4} \cos{\left(x \right)} - 120 x^{3} \sin{\left(x \right)} - 360 x^{2} \cos{\left(x \right)} + 720 x \sin{\left(x \right)} + 720 \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                     
 |                                                                                                                      
 |  6                               6               2               3             5              4                      
 | x *sin(x) dx = C + 720*cos(x) - x *cos(x) - 360*x *cos(x) - 120*x *sin(x) + 6*x *sin(x) + 30*x *cos(x) + 720*x*sin(x)
 |                                                                                                                      
/

$$\int x^{6} \sin{\left(x \right)}\, dx = C - x^{6} \cos{\left(x \right)} + 6 x^{5} \sin{\left(x \right)} + 30 x^{4} \cos{\left(x \right)} - 120 x^{3} \sin{\left(x \right)} - 360 x^{2} \cos{\left(x \right)} + 720 x \sin{\left(x \right)} + 720 \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                                                                                ___   2        ___   3     ___   6       ___   5        ___   4
                                      ___                                            ___   45*\/ 2 *pi    15*\/ 2 *pi    \/ 2 *pi    3*\/ 2 *pi    15*\/ 2 *pi 
-201792940531694400*sin(2020) + 360*\/ 2  + 67936790150008143280*cos(2020) + 90*pi*\/ 2  - ------------ - ------------ - --------- + ----------- + ------------
                                                                                                4              16           8192         1024          256

$$67936790150008143280 \cos{\left(2020 \right)} - 201792940531694400 \sin{\left(2020 \right)} - \frac{45 \sqrt{2} \pi^{2}}{4} - \frac{15 \sqrt{2} \pi^{3}}{16} - \frac{\sqrt{2} \pi^{6}}{8192} + \frac{3 \sqrt{2} \pi^{5}}{1024} + \frac{15 \sqrt{2} \pi^{4}}{256} + 90 \sqrt{2} \pi + 360 \sqrt{2}$$

                                                                                                ___   2        ___   3     ___   6       ___   5        ___   4
                                      ___                                            ___   45*\/ 2 *pi    15*\/ 2 *pi    \/ 2 *pi    3*\/ 2 *pi    15*\/ 2 *pi 
-201792940531694400*sin(2020) + 360*\/ 2  + 67936790150008143280*cos(2020) + 90*pi*\/ 2  - ------------ - ------------ - --------- + ----------- + ------------
                                                                                                4              16           8192         1024          256

-201792940531694400*sin(2020) + 360*sqrt(2) + 67936790150008143280*cos(2020) + 90*pi*sqrt(2) - 45*sqrt(2)*pi^2/4 - 15*sqrt(2)*pi^3/16 - sqrt(2)*pi^6/8192 + 3*sqrt(2)*pi^5/1024 + 15*sqrt(2)*pi^4/256

Numerical answer [src]

-1.1286603395847e+21

-1.1286603395847e+21

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

Other calculators

How to use it?

Identical expressions

Integral of x^6sinx dx

Limits of integration:

The graph:

Piecewise:

The solution