Substitution. By far the most important tool of anti-differentiation is that of applying the chain rule of differentiation backwards; that means given a function. With the substitution rule we will be able integrate a wider variety of of this is to ask yourself if you were to differentiate the integrand (we're. Using a substitution to help differentiateWe will often need to differentiate functions that are more complex than the ones that we can already do. They will simply.


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We will illustrate how this method works.

Integration by Substitution

Finally, plugging in the given function for y. Thus, we have seen in this topic differentiation by substitution the method of differentiation by substitution helps in case of difficult functions. These can be done directly term by term by inspection using the power rule backwards.

Any exponent like eax.

Calculus I - Substitution Rule for Indefinite Integrals

Again direct integration can be applied using the rule for differentiating exponents backwards. Any polynomial in sines and cosines multiplied by any exponent as in the last case. Here differentiation by substitution can use the exponential formula for sines and cosines to make this into an ugly sum of exponentials to each differentiation by substitution which case 3 applies.

Some of the exponents will be complex, but so what.


The function whose integral is the arctangent of x-a. Any rational function of x which means any function of the form where p and q are both polynomials. To do this you must be able to factor q into linear and perhaps quadratic termsdifferentiation by substitution the technique called "partial fraction expansion" discussed in a later section, and then integrate using cases 1 and 2 and perhaps 5 above.

The idea is that can be written as a sum of a polynomial and a sum over the zeroes of q x of single inverse powers and perhaps terms like that in case 5 if you don't like differentiation by substitution use complex zeroes. Any rational function of sines and cosines of x in theory anyway.

Differentiation by Substitution | [email protected]

This reduces to a rational function to which case 6 applies after the magic substitution The thought of carrying this to completion for a complicated function is too horrible for differentiation by substitution to contemplate, but it ought to work.

Here is the substitution rule in general.

Unfortunately, differentiation by substitution answer is it depends on the integral. If there is a chain rule for a derivative then there is a pretty good chance that the inside function will be the substitution that will allow us to do the integral.

Integration by Substitution

We will have to be careful however. There are times when using this general rule can get us in trouble or overly complicate the problem.

Now, with differentiation by substitution out of the way we should ask the following question. But this method only works on some integrals of course, and it may need rearranging: It is 6x, not 2x like before. Our perfect setup is gone.