Integral of 4cosx+3sinx dx

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 POST_GRBEK_SMALL_pi                        
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          2                                 
          /                                 
         |                                  
         |          (4*cos(x) + 3*sin(x)) dx
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        /                                   
POST_GRBEK_SMALL_pi                         
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         3

$$\int\limits_{\frac{POST_{GRBEK SMALL \pi}}{3}}^{\frac{POST_{GRBEK SMALL \pi}}{2}} \left(3 \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right)\, dx$$

Integral(4*cos(x) + 3*sin(x), (x, POST_GRBEK_SMALL_pi/3, POST_GRBEK_SMALL_pi/2))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3 \sin{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)}\, dx$
  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:
  
  $\int 4 \cos{\left(x \right)}\, dx = 4 \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $4 \sin{\left(x \right)}$
The result is: $4 \sin{\left(x \right)} - 3 \cos{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$4\,\sin x-3\,\cos x$$

Simplify

The answer [src]

       /POST_GRBEK_SMALL_pi\        /POST_GRBEK_SMALL_pi\        /POST_GRBEK_SMALL_pi\        /POST_GRBEK_SMALL_pi\
- 4*sin|-------------------| - 3*cos|-------------------| + 3*cos|-------------------| + 4*sin|-------------------|
       \         3         /        \         2         /        \         3         /        \         2         /

$$4\,\sin \left({{{\it POST\_GRBEK\_SMALL\_pi}}\over{2}}\right)-3\, \cos \left({{{\it POST\_GRBEK\_SMALL\_pi}}\over{2}}\right)-4\,\sin \left({{{\it POST\_GRBEK\_SMALL\_pi}}\over{3}}\right)+3\,\cos \left( {{{\it POST\_GRBEK\_SMALL\_pi}}\over{3}}\right)$$

=

       /POST_GRBEK_SMALL_pi\        /POST_GRBEK_SMALL_pi\        /POST_GRBEK_SMALL_pi\        /POST_GRBEK_SMALL_pi\
- 4*sin|-------------------| - 3*cos|-------------------| + 3*cos|-------------------| + 4*sin|-------------------|
       \         3         /        \         2         /        \         3         /        \         2         /

$$- 4 \sin{\left(\frac{POST_{GRBEK SMALL \pi}}{3} \right)} + 4 \sin{\left(\frac{POST_{GRBEK SMALL \pi}}{2} \right)} + 3 \cos{\left(\frac{POST_{GRBEK SMALL \pi}}{3} \right)} - 3 \cos{\left(\frac{POST_{GRBEK SMALL \pi}}{2} \right)}$$

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Integral of 4cosx+3sinx dx

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