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three x^ two *cos(3*x+ five)*dx
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Similar expressions
3x^2*cos(3*x-5)*dx

Integral of 3x^2cos(3x+5)*dx dx

The solution

You have entered [src]

  1                     
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 |  3*x *cos(3*x + 5) dx
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0

$$\int\limits_{0}^{1} 3 x^{2} \cos{\left(3 x + 5 \right)}\, dx$$

Integral((3*x^2)*cos(3*x + 5), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = 3 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(3 x + 5 \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 3 x + 5$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\cos{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(3 x + 5 \right)}}{3}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = 2 x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(3 x + 5 \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 3 x + 5$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\sin{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(3 x + 5 \right)}}{3}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{2 \cos{\left(3 x + 5 \right)}}{3}\right)\, dx = - \frac{2 \int \cos{\left(3 x + 5 \right)}\, dx}{3}$
1. Let $u = 3 x + 5$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\cos{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(3 x + 5 \right)}}{3}$
So, the result is: $- \frac{2 \sin{\left(3 x + 5 \right)}}{9}$
Add the constant of integration:

$x^{2} \sin{\left(3 x + 5 \right)} + \frac{2 x \cos{\left(3 x + 5 \right)}}{3} - \frac{2 \sin{\left(3 x + 5 \right)}}{9}+ \mathrm{constant}$

The answer is:

$x^{2} \sin{\left(3 x + 5 \right)} + \frac{2 x \cos{\left(3 x + 5 \right)}}{3} - \frac{2 \sin{\left(3 x + 5 \right)}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                              
 |                                                                               
 |    2                       2*sin(5 + 3*x)    2                2*x*cos(5 + 3*x)
 | 3*x *cos(3*x + 5) dx = C - -------------- + x *sin(5 + 3*x) + ----------------
 |                                  9                                   3        
/

$$\int 3 x^{2} \cos{\left(3 x + 5 \right)}\, dx = C + x^{2} \sin{\left(3 x + 5 \right)} + \frac{2 x \cos{\left(3 x + 5 \right)}}{3} - \frac{2 \sin{\left(3 x + 5 \right)}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2*cos(8)   2*sin(5)   7*sin(8)
-------- + -------- + --------
   3          9          9

$$\frac{2 \sin{\left(5 \right)}}{9} + \frac{2 \cos{\left(8 \right)}}{3} + \frac{7 \sin{\left(8 \right)}}{9}$$

2*cos(8)   2*sin(5)   7*sin(8)
-------- + -------- + --------
   3          9          9

$$\frac{2 \sin{\left(5 \right)}}{9} + \frac{2 \cos{\left(8 \right)}}{3} + \frac{7 \sin{\left(8 \right)}}{9}$$

2*cos(8)/3 + 2*sin(5)/9 + 7*sin(8)/9

Numerical answer [src]

0.459406552687302

0.459406552687302

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

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Identical expressions

Similar expressions

Integral of 3x^2cos(3x+5)*dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 3x^2*cos(3*x+5)*dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of 3x^2cos(3x+5)*dx dx