Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sin²xcos²x
Integral of 1/(x(x+2))
Integral of (1+2x^2)/(x^2(1+x^2))
Integral of -15x^2
Identical expressions
(sqrt(two)*x- three)*cos2x
( square root of (2) multiply by x minus 3) multiply by co sinus of e of 2x
( square root of (two) multiply by x minus three) multiply by co sinus of e of 2x
(√(2)*x-3)*cos2x
(sqrt(2)x-3)cos2x
sqrt2x-3cos2x
(sqrt(2)*x-3)*cos2xdx
Similar expressions
((sqrt(2x)-3)*cos2x)
(sqrt(2)*x+3)*cos2x

Integral of (sqrt(2)x-3)cos2x dx

The solution

You have entered [src]

  1                          
  /                          
 |                           
 |  /  ___      \            
 |  \\/ 2 *x - 3/*cos(2*x) dx
 |                           
/                            
0

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

Integral((sqrt(2)*x - 3)*cos(2*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{2} x$ .
  
  Then let $du = \sqrt{2} dx$ and substitute $du$ :
  
  $\int \left(\frac{\sqrt{2} u \cos{\left(\sqrt{2} u \right)}}{2} - \frac{3 \sqrt{2} \cos{\left(\sqrt{2} u \right)}}{2}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sqrt{2} u \cos{\left(\sqrt{2} u \right)}}{2}\, du = \frac{\sqrt{2} \int u \cos{\left(\sqrt{2} u \right)}\, du}{2}$
      
      Let $u = \sqrt{2} u$ .
      
      Then let $du = \sqrt{2} du$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{u \cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u \cos{\left(u \right)}\, du = \frac{\int u \cos{\left(u \right)}\, du}{2}$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = \cos{\left(u \right)}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $\frac{u \sin{\left(u \right)}}{2} + \frac{\cos{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sqrt{2} u \sin{\left(\sqrt{2} u \right)}}{2} + \frac{\cos{\left(\sqrt{2} u \right)}}{2}$
      
      So, the result is: $\frac{\sqrt{2} \left(\frac{\sqrt{2} u \sin{\left(\sqrt{2} u \right)}}{2} + \frac{\cos{\left(\sqrt{2} u \right)}}{2}\right)}{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{3 \sqrt{2} \cos{\left(\sqrt{2} u \right)}}{2}\right)\, du = - \frac{3 \sqrt{2} \int \cos{\left(\sqrt{2} u \right)}\, du}{2}$
      
      Let $u = \sqrt{2} u$ .
      
      Then let $du = \sqrt{2} du$ and substitute $\frac{\sqrt{2} du}{2}$ :
      
      $\int \frac{\sqrt{2} \cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\sqrt{2} \int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sqrt{2} \sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sqrt{2} \sin{\left(\sqrt{2} u \right)}}{2}$
      
      So, the result is: $- \frac{3 \sin{\left(\sqrt{2} u \right)}}{2}$
    The result is: $\frac{\sqrt{2} \left(\frac{\sqrt{2} u \sin{\left(\sqrt{2} u \right)}}{2} + \frac{\cos{\left(\sqrt{2} u \right)}}{2}\right)}{2} - \frac{3 \sin{\left(\sqrt{2} u \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\sqrt{2} \left(x \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{2}\right)}{2} - \frac{3 \sin{\left(2 x \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\left(\sqrt{2} x - 3\right) \cos{\left(2 x \right)} = \sqrt{2} x \cos{\left(2 x \right)} - 3 \cos{\left(2 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sqrt{2} x \cos{\left(2 x \right)}\, dx = \sqrt{2} \int x \cos{\left(2 x \right)}\, dx$
    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 $\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(2 x \right)}}{2}\, dx = \frac{\int \sin{\left(2 x \right)}\, dx}{2}$
      
      There are multiple ways to do this integral.
      
      Method #1
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\sin{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(2 x \right)}}{2}$
      
      Method #2
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{2}{\left(x \right)}}{2}$
      
      So, the result is: $- \cos^{2}{\left(x \right)}$
      
      So, the result is: $- \frac{\cos{\left(2 x \right)}}{4}$
    So, the result is: $\sqrt{2} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}\right)$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \cos{\left(2 x \right)}\right)\, dx = - 3 \int \cos{\left(2 x \right)}\, dx$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $- \frac{3 \sin{\left(2 x \right)}}{2}$
  The result is: $\sqrt{2} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}\right) - \frac{3 \sin{\left(2 x \right)}}{2}$
Method #3
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \sqrt{2} x - 3$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \sqrt{2}$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = 2 x$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\cos{\left(u \right)}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(2 x \right)}}{2}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sqrt{2} \sin{\left(2 x \right)}}{2}\, dx = \frac{\sqrt{2} \int \sin{\left(2 x \right)}\, dx}{2}$
  1. Let $u = 2 x$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\sin{\left(u \right)}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $- \frac{\cos{\left(2 x \right)}}{2}$
  So, the result is: $- \frac{\sqrt{2} \cos{\left(2 x \right)}}{4}$
Method #4
1. Rewrite the integrand:
  
  $\left(\sqrt{2} x - 3\right) \cos{\left(2 x \right)} = \sqrt{2} x \cos{\left(2 x \right)} - 3 \cos{\left(2 x \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sqrt{2} x \cos{\left(2 x \right)}\, dx = \sqrt{2} \int x \cos{\left(2 x \right)}\, dx$
    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 $\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(2 x \right)}}{2}\, dx = \frac{\int \sin{\left(2 x \right)}\, dx}{2}$
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\sin{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(2 x \right)}}{2}$
      
      So, the result is: $- \frac{\cos{\left(2 x \right)}}{4}$
    So, the result is: $\sqrt{2} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}\right)$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 \cos{\left(2 x \right)}\right)\, dx = - 3 \int \cos{\left(2 x \right)}\, dx$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $- \frac{3 \sin{\left(2 x \right)}}{2}$
  The result is: $\sqrt{2} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}\right) - \frac{3 \sin{\left(2 x \right)}}{2}$
Now simplify:

$\frac{\sqrt{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right)}{4} - \frac{3 \sin{\left(2 x \right)}}{2}$
Add the constant of integration:

$\frac{\sqrt{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right)}{4} - \frac{3 \sin{\left(2 x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right)}{4} - \frac{3 \sin{\left(2 x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               ___ /cos(2*x)             \
 |                                              \/ 2 *|-------- + x*sin(2*x)|
 | /  ___      \                   3*sin(2*x)         \   2                 /
 | \\/ 2 *x - 3/*cos(2*x) dx = C - ---------- + -----------------------------
 |                                     2                      2              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

               ___     ___            ___       
  3*sin(2)   \/ 2    \/ 2 *sin(2)   \/ 2 *cos(2)
- -------- - ----- + ------------ + ------------
     2         4          2              4

$$- \frac{3 \sin{\left(2 \right)}}{2} - \frac{\sqrt{2}}{4} + \frac{\sqrt{2} \cos{\left(2 \right)}}{4} + \frac{\sqrt{2} \sin{\left(2 \right)}}{2}$$

               ___     ___            ___       
  3*sin(2)   \/ 2    \/ 2 *sin(2)   \/ 2 *cos(2)
- -------- - ----- + ------------ + ------------
     2         4          2              4

$$- \frac{3 \sin{\left(2 \right)}}{2} - \frac{\sqrt{2}}{4} + \frac{\sqrt{2} \cos{\left(2 \right)}}{4} + \frac{\sqrt{2} \sin{\left(2 \right)}}{2}$$

-3*sin(2)/2 - sqrt(2)/4 + sqrt(2)*sin(2)/2 + sqrt(2)*cos(2)/4

Numerical answer [src]

-1.22165927925379

-1.22165927925379

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (sqrt(2)x-3)cos2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2

Method #3

Method #4

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (sqrt(2)*x-3)*cos2x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2

Method #3

Method #4

Integral of (sqrt(2)x-3)cos2x dx