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Solving integrals/ cosx*sin(4sinx-1)dx

Integral of cosx*sin(4sinx-1)dx dx

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = 4 \sin{\left(x \right)} - 1$ .

Then let $du = 4 \cos{\left(x \right)} dx$ and substitute $\frac{du}{4}$ :

$\int \frac{\sin{\left(u \right)}}{16}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sin{\left(u \right)}}{4}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
  So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
Now substitute $u$ back in:

$- \frac{\cos{\left(4 \sin{\left(x \right)} - 1 \right)}}{4}$
Now simplify:

$- \frac{\cos{\left(4 \sin{\left(x \right)} - 1 \right)}}{4}$
Add the constant of integration:

$- \frac{\cos{\left(4 \sin{\left(x \right)} - 1 \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(4 \sin{\left(x \right)} - 1 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                     
 |                                     cos(4*sin(x) - 1)
 | cos(x)*sin(4*sin(x) - 1)*1 dx = C - -----------------
 |                                             4        
/

$$\int \cos{\left(x \right)} \sin{\left(4 \sin{\left(x \right)} - 1 \right)} 1\, dx = C - \frac{\cos{\left(4 \sin{\left(x \right)} - 1 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  cos(1 - 4*sin(1))   cos(1)
- ----------------- + ------
          4             4

$$\frac{\cos{\left(1 \right)}}{4} - \frac{\cos{\left(1 - 4 \sin{\left(1 \right)} \right)}}{4}$$

  cos(1 - 4*sin(1))   cos(1)
- ----------------- + ------
          4             4

$$\frac{\cos{\left(1 \right)}}{4} - \frac{\cos{\left(1 - 4 \sin{\left(1 \right)} \right)}}{4}$$

Упростить

Numerical answer [src]

0.31355681542951

0.31355681542951

The graph

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

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Identical expressions

Similar expressions

Integral of cosx*sin(4sinx-1)dx dx

Limits of integration:

The graph:

Piecewise:

The solution