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Identical expressions
4cos2x-3sinx
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Similar expressions
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Integral of 4cos2x-3sinx dx

The solution

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  1                           
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 |  (4*cos(2*x) - 3*sin(x)) dx
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0

$$\int\limits_{0}^{1} \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(2 x \right)}\right)\, dx$$

Integral(4*cos(2*x) - 3*sin(x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 3 \sin{\left(x \right)}\right)\, dx = - 3 \int \sin{\left(x \right)}\, dx$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $3 \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 \cos{\left(2 x \right)}\, dx = 4 \int \cos{\left(2 x \right)}\, dx$
  1. Let $u = 2 x$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{\cos{\left(u \right)}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(2 x \right)}}{2}$
  So, the result is: $2 \sin{\left(2 x \right)}$
The result is: $2 \sin{\left(2 x \right)} + 3 \cos{\left(x \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                      
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 | (4*cos(2*x) - 3*sin(x)) dx = C + 2*sin(2*x) + 3*cos(x)
 |                                                       
/

$$\int \left(- 3 \sin{\left(x \right)} + 4 \cos{\left(2 x \right)}\right)\, dx = C + 2 \sin{\left(2 x \right)} + 3 \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-3 + 2*sin(2) + 3*cos(1)

$$-3 + 3 \cos{\left(1 \right)} + 2 \sin{\left(2 \right)}$$

-3 + 2*sin(2) + 3*cos(1)

$$-3 + 3 \cos{\left(1 \right)} + 2 \sin{\left(2 \right)}$$

-3 + 2*sin(2) + 3*cos(1)

Numerical answer [src]

0.439501771255783

0.439501771255783

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

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Identical expressions

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Integral of 4cos2x-3sinx dx

Limits of integration:

The graph:

Piecewise:

The solution