Could someone explain how my teacher got from step 1 to step 2. I am very new to integrals so please don't skip key things and assume I know them.
The integral has X at the top of the "S" and 0 at the bottom of the "S"
Step 1: [dX/(A-BX)] into Step 2: (-1/B)[ln(A-BX)-ln(A)]
I don't get how he went from step 1 to step 2. THANK YOU!
The integral has X at the top of the "S" and 0 at the bottom of the "S"
Step 1: [dX/(A-BX)] into Step 2: (-1/B)[ln(A-BX)-ln(A)]
I don't get how he went from step 1 to step 2. THANK YOU!
-
ok i see how to help
first you want to use substitution
the part on the bottom is a bit complex ... lets change it
u = A - BX
now take the derivative of this
du/dx = -B
-du/B = dx
==========
thats step one
now we can change the integral
substitute the (-du/B) for the dx
and u for the (A - BX)
integral -(du/B)/u
integral -(1/B)du/u
so we can take the -(1/B) out of the integral
-(1/B) integral du/u
the integral of du/u = ln u
this you just have to know
because d(lnu)/du = 1/u
I cant hellp you there you just have to know it
so we have the solution
-(1/B) integral du/u
-(1/B) integral of du/u = -(1/B)ln u
now we change the ln u back to the value it was substituted for
u = A - BX
-(1/B) ln(A-BX)
now remember this has to be evaluated at the two points we were given
the point on the top of the S or integral sign was X = x
the point on the bottom o f the integral sign was X = 0
I put the X for the variable in the integral and the little x for the value of the endpoint on the integral sign
when you evaluate at the end points you do the top value first then subtract it from the bottom point..
...........................|X = x
-(1/B) ln(A-BX) |
..........................| X = 0
becomes
-(1/B)* (ln(A -Bx) - ln(A - B*0))
obviously 0*B = 0
-(1/B)*(ln(A - Bx) - ln(A))
that is how you get that answer
I hope I helped its hard to show the integral sign in this text mode..
first you want to use substitution
the part on the bottom is a bit complex ... lets change it
u = A - BX
now take the derivative of this
du/dx = -B
-du/B = dx
==========
thats step one
now we can change the integral
substitute the (-du/B) for the dx
and u for the (A - BX)
integral -(du/B)/u
integral -(1/B)du/u
so we can take the -(1/B) out of the integral
-(1/B) integral du/u
the integral of du/u = ln u
this you just have to know
because d(lnu)/du = 1/u
I cant hellp you there you just have to know it
so we have the solution
-(1/B) integral du/u
-(1/B) integral of du/u = -(1/B)ln u
now we change the ln u back to the value it was substituted for
u = A - BX
-(1/B) ln(A-BX)
now remember this has to be evaluated at the two points we were given
the point on the top of the S or integral sign was X = x
the point on the bottom o f the integral sign was X = 0
I put the X for the variable in the integral and the little x for the value of the endpoint on the integral sign
when you evaluate at the end points you do the top value first then subtract it from the bottom point..
...........................|X = x
-(1/B) ln(A-BX) |
..........................| X = 0
becomes
-(1/B)* (ln(A -Bx) - ln(A - B*0))
obviously 0*B = 0
-(1/B)*(ln(A - Bx) - ln(A))
that is how you get that answer
I hope I helped its hard to show the integral sign in this text mode..