Applied Calculus; An Introductory Textbook
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To quickly assess the difficulty of the text, read a short excerpt:
Complicated expressions to be integrated can often be simplified by expressing them in terms of a new variable.
Suppose / d){x)dx is required.
J a We may suppose jjc to be a function of a new vari- able 0, provided that the new function of z chosen can assume all the values of x required in the integra- tion.
For instance, if I xl^dx is required, we could not put x = sin z because sin z cannot be g^reater than i.
But we could put x = tan z, because a "sr" can be found to give tan z equal to any... number between i and 3.
It would not simplify the work, in this instance, to do this — the example is merely intended to show why certain functions are admis- sible while others are not.
INTEGRATION 157 Let I = \f{x)dx.
Then -t- = fix) = ^(z) if x is a function of z.
Let a; be a differentiable function of z, say 9^(0), ^, dl d\ dx . , V = f^^)fz :. I ^ i[,riz)^]dz (4) This is the transformation formula for substitution.
EXAMPLES (i) Evaluate I sin {2x)dx.
Put 2X = s.
, dx I .'. I = jsinz X i k dz = ^ I sin zdz = — ^ cos z = — ^ cos {zx) + c* (ii) Evaluate I sin {mx + w)
What to read after Applied Calculus; An Introductory Textbook?
You can find similar books in the "Read Also" column, or choose other free books by Frederick Francis Percival Bisacre to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Complicated expressions to be integrated can often be simplified by expressing them in terms of a new variable.
Suppose / d){x)dx is required.
J a We may suppose jjc to be a function of a new vari- able 0, provided that the new function of z chosen can assume all the values of x required in the integra- tion.
For instance, if I xl^dx is required, we could not put x = sin z because sin z cannot be g^reater than i.
But we could put x = tan z, because a "sr" can be found to give tan z equal to any... number between i and 3.
It would not simplify the work, in this instance, to do this — the example is merely intended to show why certain functions are admis- sible while others are not.
INTEGRATION 157 Let I = \f{x)dx.
Then -t- = fix) = ^(z) if x is a function of z.
Let a; be a differentiable function of z, say 9^(0), ^, dl d\ dx . , V = f^^)fz :. I ^ i[,riz)^]dz (4) This is the transformation formula for substitution.
EXAMPLES (i) Evaluate I sin {2x)dx.
Put 2X = s.
, dx I .'. I = jsinz X i k dz = ^ I sin zdz = — ^ cos z = — ^ cos {zx) + c* (ii) Evaluate I sin {mx + w)
What to read after Applied Calculus; An Introductory Textbook?
You can find similar books in the "Read Also" column, or choose other free books by Frederick Francis Percival Bisacre to read onlineMoreLess
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