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Integral of d{x}:
Integral of x^(-2)
Integral of sin(3x)
Integral of logx
Integral of x^2/(1+x)
Identical expressions
(sin(three x))/((cos(3x)^(four \3)))
( sinus of (3x)) divide by (( co sinus of e of (3x) to the power of (4\3)))
( sinus of (three x)) divide by (( co sinus of e of (3x) to the power of (four \3)))
(sin(3x))/((cos(3x)(4\3)))
sin3x/cos3x4\3
sin3x/cos3x^4\3
(sin(3x)) divide by ((cos(3x)^(4\3)))
(sin(3x))/((cos(3x)^(4\3)))dx

Solving integrals/ (sin(3x))/((cos(3x)^(4\3)))

Integral of (sin(3x))/((cos(3x)^(4\3))) dx

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = 3 x$ .

Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :

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

$\frac{1}{\sqrt[3]{\cos{\left(3 x \right)}}}$
Add the constant of integration:

$\frac{1}{\sqrt[3]{\cos{\left(3 x \right)}}}+ \mathrm{constant}$

The answer is:

$\frac{1}{\sqrt[3]{\cos{\left(3 x \right)}}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |   sin(3*x)                1      
 | ----------- dx = C + ------------
 |    4/3               3 __________
 | cos   (3*x)          \/ cos(3*x) 
 |                                  
/

$$\int \frac{\sin{\left(3 x \right)}}{\cos^{\frac{4}{3}}{\left(3 x \right)}}\, dx = C + \frac{1}{\sqrt[3]{\cos{\left(3 x \right)}}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         1               /    2/3\
oo + ---------- + oo*sign\(-3)   /
     3 ________                   
     \/ cos(3)

$$\infty + \frac{1}{\sqrt[3]{\cos{\left(3 \right)}}} + \infty \operatorname{sign}{\left(\left(-3\right)^{\frac{2}{3}} \right)}$$

         1               /    2/3\
oo + ---------- + oo*sign\(-3)   /
     3 ________                   
     \/ cos(3)

$$\infty + \frac{1}{\sqrt[3]{\cos{\left(3 \right)}}} + \infty \operatorname{sign}{\left(\left(-3\right)^{\frac{2}{3}} \right)}$$

oo + cos(3)^(-1/3) + oo*sign((-3)^(2/3))

Numerical answer [src]

(-14.0822511887913 + 29.1794086862673j)

(-14.0822511887913 + 29.1794086862673j)

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

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Identical expressions

Integral of (sin(3x))/((cos(3x)^(4\3))) dx

Limits of integration:

The graph:

Piecewise:

The solution