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release

Evaluate integrals

所有的页面崩溃

Syntax

release(expr)

Description

example

release(expr)evaluates the integrals in the expressionexpr. Thereleasefunction ignores the'Hold'option in theintfunction when the integrals are defined.

Examples

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Open Live Script

Define a symbolic call to an integral ∫ cos ( x ) dx without evaluating it. Set the'Hold'option to true when defining the integral using theintfunction.

symsxF = int(cos(x),'Hold',真正的)
F =

                      

                       
                        
                         
                          
                           
                            ∫
                           
                           
                            
                             
                              
                               cos
                             
                             
                              
                               (
                              
                               
                                
                                 x
                               
                              
                              
                               )
                             
                            
                           
                           
                            d
                           
                            
                             x
                           
                          
                         
                         
                          int(cos(x), x, 'Hold = TRUE', true)

Usereleaseto evaluate the integral by ignoring the'Hold'option.

G =释放(F)
G =
                      
                       
                        
                         
                          
                           sin
                         
                         
                          
                           (
                          
                           
                            
                             x
                           
                          
                          
                           )
                         
                        
                        
                         sin(x)
Open Live Script

Find the integral of ∫ x e x dx .

Define the integral without evaluating it by setting the'Hold'option totrue.

symsxg(y)F = int(x*exp(x),'Hold',真正的)
F =

                      

                       
                        
                         
                          
                           
                            ∫
                           
                           
                            
                             
                              
                               x
                              
                              
                              
                               
                                
                                 e
                               
                               
                                
                                 x
                               
                              
                             
                            
                           
                           
                            d
                           
                            
                             x
                           
                          
                         
                         
                          int(x*exp(x), x, 'Hold = TRUE', true)

You can apply integration by parts toFby using theintegrateByPartsfunction. Useexp(x)as the differential to be integrated.

G = integrateByParts(F,exp(x))
G =

                      

                       
                        
                         
                          
                           
                            
                             
                              x
                             
                             
                             
                              
                               
                                e
                              
                              
                               
                                x
                              
                             
                            
                           
                           
                            -
                           
                            
                             ∫
                            
                            
                             
                              
                               
                                e
                              
                              
                               
                                x
                              
                             
                            
                            
                             d
                            
                             
                              x
                            
                           
                          
                         
                         
                          x*exp(x) - int(exp(x), x, 'Hold = TRUE', true)

To evaluate the integral inG, use thereleasefunction to ignore the'Hold'option.

Gcalc = release(G)
Gcalc =
                      
                       
                        
                         
                          
                           
                            
                             x
                            
                            
                            
                             
                              
                               e
                             
                             
                              
                               x
                             
                            
                           
                          
                          
                           -
                          
                           
                            
                             e
                           
                           
                            
                             x
                           
                          
                         
                        
                        
                         x*exp(x) - exp(x)

Compare the result to the integration result returned byintwithout setting the'Hold'option.

Fcalc = int(x*exp(x))
Fcalc =
                      
                       
                        
                         
                          
                           
                            
                             e
                           
                           
                            
                             x
                           
                          
                          
                          
                          
                           
                            
                             
                              
                               x
                              
                               -
                              
                               1
                             
                            
                           
                          
                         
                        
                        
                         exp(x)*(x - 1)
Open Live Script

Find the integral of ∫ cos ( log ( x ) ) dx using integration by substitution.

Define the integral without evaluating it by setting the'Hold'option totrue.

symsxtF = int(cos(log(x)),'Hold',真正的)
F =

                      

                       
                        
                         
                          
                           
                            ∫
                           
                           
                            
                             
                              
                               cos
                             
                             
                              
                               (
                              
                               
                                
                                 
                                  
                                   log
                                 
                                 
                                  
                                   (
                                  
                                   
                                    
                                     x
                                   
                                  
                                  
                                   )
                                 
                                
                               
                              
                              
                               )
                             
                            
                           
                           
                            d
                           
                            
                             x
                           
                          
                         
                         
                          int(cos(log(x)), x, 'Hold = TRUE', true)

Substitute the expressionlog(x)witht.

G = changeIntegrationVariable(F,log(x),t)
G =

                      

                       
                        
                         
                          
                           
                            ∫
                           
                           
                            
                             
                              
                               
                                
                                 e
                               
                               
                                
                                 t
                               
                              
                              
                              
                              
                               
                                
                                 cos
                               
                               
                                
                                 (
                                
                                 
                                  
                                   t
                                 
                                
                                
                                 )
                               
                              
                             
                            
                           
                           
                            d
                           
                            
                             t
                           
                          
                         
                         
                          int(exp(t)*cos(t), t, 'Hold = TRUE', true)

To evaluate the integral inG, use thereleasefunction to ignore the'Hold'option.

H = release(G)
H =

                      

                       
                        
                         
                          
                           
                            
                             
                              
                               
                                
                                 e
                               
                               
                                
                                 t
                               
                              
                              
                              
                              
                               
                                
                                 
                                  
                                   
                                    
                                     cos
                                   
                                   
                                    
                                     (
                                    
                                     
                                      
                                       t
                                     
                                    
                                    
                                     )
                                   
                                  
                                  
                                   +
                                  
                                   
                                    
                                     sin
                                   
                                   
                                    
                                     (
                                    
                                     
                                      
                                       t
                                     
                                    
                                    
                                     )
                                   
                                  
                                 
                                
                               
                              
                             
                            
                           
                           
                            
                             2
                           
                          
                         
                         
                          (exp(t)*(cos(t) + sin(t)))/2

Restorelog(x)in place oft.

H = simplify(subs(H,t,log(x)))
H =

                      

                       
                        
                         
                          
                           
                            
                             
                              
                               
                                
                                 2
                               
                              
                              
                              
                              
                               x
                              
                              
                              
                               
                                
                                 sin
                               
                               
                                
                                 (
                                
                                 
                                  
                                   
                                    
                                     
                                      
                                       π
                                     
                                     
                                      
                                       4
                                     
                                    
                                    
                                     +
                                    
                                     
                                      
                                       log
                                     
                                     
                                      
                                       (
                                      
                                       
                                        
                                         x
                                       
                                      
                                      
                                       )
                                     
                                    
                                   
                                  
                                 
                                
                                
                                 )
                               
                              
                             
                            
                           
                           
                            
                             2
                           
                          
                         
                         
                          (sqrt(sym(2))*x*sin(sym(pi)/4 + log(x)))/2

Compare the result to the integration result returned byintwithout setting the'Hold'option totrue.

Fcalc = int(cos(log(x)))
Fcalc =

                      

                       
                        
                         
                          
                           
                            
                             
                              
                               
                                
                                 2
                               
                              
                              
                              
                              
                               x
                              
                              
                              
                               
                                
                                 sin
                               
                               
                                
                                 (
                                
                                 
                                  
                                   
                                    
                                     
                                      
                                       π
                                     
                                     
                                      
                                       4
                                     
                                    
                                    
                                     +
                                    
                                     
                                      
                                       log
                                     
                                     
                                      
                                       (
                                      
                                       
                                        
                                         x
                                       
                                      
                                      
                                       )
                                     
                                    
                                   
                                  
                                 
                                
                                
                                 )
                               
                              
                             
                            
                           
                           
                            
                             2
                           
                          
                         
                         
                          (sqrt(sym(2))*x*sin(sym(pi)/4 + log(x)))/2

Input Arguments

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Expression containing integrals, specified as a symbolic expression, function, vector, or matrix.

See Also

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Introduced in R2019b

Symbolic Math Toolbox Documentation

金宝app

Mathematical Modeling with Symbolic Math Toolbox

Mathematical Modeling with Symbolic Math Toolbox

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