what is the integral of 1dx/(50-10x) is it -1/10ln(50-10x) +c or -1/10ln(5-x) +c

Question

∫dx/(50-10x)=?

 

Does factoring out 1/10 change the answer.

Answer 1

∫ dx/(50 - 10x)  <------ Let's use u-substitution

 

u =  50 - 10x 

du = -10 dx  <----which implies ---> du/(-10) = dx

 

∫ dx/(u)

 

∫ ( du/(-10) )/(u) <----- let's rearrange  

 

∫ ( du/(-10) ) * (1/u)

 

∫ ( du/(u * -10) ) 

 

∫ (-1/10) * (1/u) * du 

 

(-1/10) * ln( u ) + C 

 

(-1/10) * ln( 50 - 10x ) + C 

 

 

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Answer 2

If you were rewrite the integral by factoring out the 1/10 and then use the substitution method that Mohammad provided to integrate, you will get

 

(-1/10)ln(5 - x) + C

 

which appears a bit different from what Mohammad got, but are actually the same.  When taking the derivative of this integral, you will get the original expression you need to integrate.

 

 

Taking the derivative of  both    (-1/10)ln(5 - x)+C      and        (-1/10)ln(50 - 10x)+C     will result in     1 / (50 - 10x).  This is how you would check your integral to see if you are correct.

 

To completely answer your question, factoring out a 1/10 will NOT change your answer.