Can you please explain this equality? ∫ r σ 2 exp ⁡ ( − r 2 2 σ 2 ) d r = − exp ⁡ ( − r 2 2 σ 2 )

Question

Can you please explain this equality?  ∫      r          σ      2        exp  ⁡      (                  −                  r          2                            2                  σ          2                      )      d  r  =  −  exp  ⁡      (                  −                  r          2                            2                  σ          2                      )

Ramiro Sosa · Accepted Answer

To check that an indefinite integral is equal to a function  ∫  f  (  x  )    d  x  =  F  (  x  )    (  +  C  )you can simply check that       F    ′    (  x  )  =  f  (  x  ). In your example you check that      d          d      r        [  −  exp  ⁡      (                  −                  r          2                            2                  σ          2                      )    ]  =  −  exp  ⁡      (                  −                  r          2                            2                  σ          2                      )        d          d      r                  −              r        2                    2              σ        2                =  exp  ⁡      (                  −                  r          2                            2                  σ          2                      )        r          σ      2      If you don&#039;t like this, then you can also find the integral using the substitution method with             u      =                        −                      r            2                                    2                      σ            2                              . So then                    d        u                            =                              −            2            r                                2                          σ              2                                      d        r                ⇒                            −        d        u                            =                  r                      σ            2                          d        r            So the integral becomes  ∫  −      e    u      d  u  .

Can you please explain this equality? int r/theta^2 exp(-r^2/(2theta^2))dr=-exp(-r^2/2theta^2)

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