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Solving integrals/ (3-4sinx)^1/3*cos

Integral of (3-4sinx)^1/3*cos dx

The solution

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  1                           
  /                           
 |                            
 |  3 ______________          
 |  \/ 3 - 4*sin(x) *cos(x) dx
 |                            
/                             
0

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

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

Detail solution

Let $u = 3 - 4 \sin{\left(x \right)}$ .

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

$\int \frac{\sqrt[3]{u}}{16}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sqrt[3]{u}}{4}\right)\, du = - \frac{\int \sqrt[3]{u}\, du}{4}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt[3]{u}\, du = \frac{3 u^{\frac{4}{3}}}{4}$
  So, the result is: $- \frac{3 u^{\frac{4}{3}}}{16}$
Now substitute $u$ back in:

$- \frac{3 \left(3 - 4 \sin{\left(x \right)}\right)^{\frac{4}{3}}}{16}$
Now simplify:

$- \frac{3 \left(3 - 4 \sin{\left(x \right)}\right)^{\frac{4}{3}}}{16}$
Add the constant of integration:

$- \frac{3 \left(3 - 4 \sin{\left(x \right)}\right)^{\frac{4}{3}}}{16}+ \mathrm{constant}$

The answer is:

$- \frac{3 \left(3 - 4 \sin{\left(x \right)}\right)^{\frac{4}{3}}}{16}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
 |                                                  4/3
 | 3 ______________                 3*(3 - 4*sin(x))   
 | \/ 3 - 4*sin(x) *cos(x) dx = C - -------------------
 |                                           16        
/

$$\int \sqrt[3]{3 - 4 \sin{\left(x \right)}} \cos{\left(x \right)}\, dx = C - \frac{3 \left(3 - 4 \sin{\left(x \right)}\right)^{\frac{4}{3}}}{16}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    3 ______________     3 ___     3 ______________       
  9*\/ 3 - 4*sin(1)    9*\/ 3    3*\/ 3 - 4*sin(1) *sin(1)
- ------------------ + ------- + -------------------------
          16              16                 4

$$\frac{9 \cdot \sqrt[3]{3}}{16} - \frac{9 \sqrt[3]{3 - 4 \sin{\left(1 \right)}}}{16} + \frac{3 \sqrt[3]{3 - 4 \sin{\left(1 \right)}} \sin{\left(1 \right)}}{4}$$

    3 ______________     3 ___     3 ______________       
  9*\/ 3 - 4*sin(1)    9*\/ 3    3*\/ 3 - 4*sin(1) *sin(1)
- ------------------ + ------- + -------------------------
          16              16                 4

$$\frac{9 \cdot \sqrt[3]{3}}{16} - \frac{9 \sqrt[3]{3 - 4 \sin{\left(1 \right)}}}{16} + \frac{3 \sqrt[3]{3 - 4 \sin{\left(1 \right)}} \sin{\left(1 \right)}}{4}$$

Упростить

Numerical answer [src]

(0.835914479962129 + 0.0425450463965593j)

(0.835914479962129 + 0.0425450463965593j)

The graph

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

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Integral of (3-4sinx)^1/3*cos dx

Limits of integration:

The graph:

Piecewise:

The solution