We can use integration by substitution to undo differentiation that has been done using the chain rule. It gives us a way to turn some complicated, scary-looking integrals into ones that are easy to deal with.

To use this technique, we need to be able to write our integral in the form shown below:. So, this example is in good shape and ready for us to apply our brand new, fancy-shmancy integration technique. So, what do we do now? I'm glad you asked. We do our substitution. So, here's a question for you Now it's time to put our theory into practice. Let's see if we can integrate our real-life example.

Now let's try some slightly harder examples - ones that aren't quite so nicely set up for substitution. This integrand wasn't quite so thoughtful in the way it expressed itself.

There's no need to panic. The rules of integration get us out of trouble here. All you need to do is pull a 4 out the front of the integral, like this:. If you're not sure why this is OK, check out the "multiplication by a constant" rule in the rules of integration article.

Don't worry, we'll still be here when you get back! Here we go:. Let's look at one last example. At first, it doesn't look like substitution will work on this one, but it will. We now have something else to add to our list of integration party-tricks.

Integration by substitution helps us to turn mean, nasty, complicated integrals into nice, friendly, cuddly integrals that we can evaluate. Remember the steps:. Author: Subject Coach Added on: 23rd Nov Tutorial Feedback Tweet. Integration by Substitution. Integration by Substitution We can use integration by substitution to undo differentiation that has been done using the chain rule.

Environment It is considered a good practice to take notes and revise what you learnt and practice it. You must be logged in as Student to ask a Question. Get FREE educational material sent directly to your inbox. Tutorial Feedback.We motivate this section with an example. It is:. This section explores integration by substitution. It allows us to "undo the Chain Rule. We'll formally establish later how this is done.

We wish to make this simpler; we do so through a substitution. One might well look at this and think "I sort of followed how that worked, but I could never come up with that on my own," but the process is learnable. This section contains numerous examples through which the reader will gain understanding and mathematical maturity enabling them to regard substitution as a natural tool when evaluating integrals.

We stated before that integration by substitution "undoes" the Chain Rule. The point of substitution is to make the integration step easy. The "work" involved is making the proper substitution. There is not a step-by-step process that one can memorize; rather, experience will be one's guide.

To gain experience, we now embark on many examples. This is not always a good choice, but it is often the best place to start. We can check our work by evaluating the derivative of the right hand side. We can now substitute. The previous example exhibited a common, and simple, type of substitution. When the inside function is linear, the resulting integration is very predictable, outlined here.

### 𝘶-substitution intro

Our next example can use Key Idea 10, but we will only employ it after going through all of the steps. We can now evaluate the integral through substitution. One may want to continue writing out all the steps until they are comfortable with this particular shortcut. Not all integrals that benefit from substitution have a clear "inside" function.

Several of the following examples will demonstrate ways in which this occurs. There is not a composition of function here to exploit; rather, just a product of functions. The substitution becomes very straightforward:. Our examples so far have required "basic substitution. But at this stage, we have:. Checking your work is always a good idea. In this particular case, some algebra will be needed to make one's answer match the integrand in the original problem.

This is another example where there does not seem to be an obvious composition of functions. The final answer is interesting; the natural log of the natural log. Take the derivative to confirm this answer is indeed correct. Section 6.If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

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## Integration by Substitution

Current timeTotal duration Google Classroom Facebook Twitter. Video transcript Let's say that we have the indefinite integral, and the function is 3x squared plus 2x times e to x to the third plus x squared dx. So how would we go about solving this? So first when you look at it, it seems like a really complicated integral. We have this polynomial right over here being multiplied by this exponential expression, and over here in the exponent, we essentially have another polynomial.

It seems kind of crazy. And the key intuition here, the key insight is that you might want to use a technique here called u-substitution. And I'll tell you in a second how I would recognize that we have to use u-substitution. And then over time, you might even be able to do this type of thing in your head. And the chain rule-- I'll go in more depth in another video, where I really talk about that intuition.

But the way I would think about it is, well, I have this crazy exponent right over here. I have the x to the third plus x squared, and this thing right over here happens to be the derivative of x to the third plus x squared. The derivative of x to the third is 3x squared, derivative of x squared is 2x, which is a huge clue to me that I could use u-substitution. So what I do here is this thing, or this little expression here, where I also see its derivative being multiplied, I can set that equal to u.

So I can say u is equal to x to the third plus x squared. Now, what is going to be the derivative of u with respect to x? Well, we've done this multiple times. It's going to be 3x squared plus 2x. And now we can write this in differential form. And du dx, this isn't really a fraction of the differential of du divided by differential of dx.

It really is a form of notation, but it is often useful to kind of pretend that it is a fraction, and you could kind of view this if you wanted to just get a du, if you just wanted to get a differential form over here, how much does u change for a given change in x? You could multiply both sides times a dx. So both sides times a dx.

And so if we were to pretend that they were fractions, and it will give you the correct differential form, you're going to be left with du is equal to 3x squared plus 2x dx.

Now why is this over here? Why did I go to the trouble of doing that?All of these look considerably more difficult than the first set. Here is the substitution rule in general. A natural question at this stage is how to identify the correct substitution. Unfortunately, the answer is it depends on the integral. With the integral above we can quickly recognize that we know how to integrate. 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.

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 that out of the way we should ask the following question. How, do we know if we got the correct substitution? Again, it cannot be stressed enough at this point that the only way to really learn how to do substitutions is to just work lots of problems. So, as with the first example we worked the stuff in front of the cosine appears exactly in the differential.

The integral is then. Again, it looks like we have an exponential function with an inside function i. Now, with the exception of the 3 the stuff in front of the exponential appears exactly in the differential.

Okay, now we have a small problem. This is not really the problem it might appear to be at first. We will simply rewrite the differential as follows. Note that in most problems when we pick up a constant as we did in this example we will generally factor it out of the integral in the same step that we substitute it in.

Upon doing this we can see that the substitution is. In the previous set of examples the substitution was generally pretty clear. There are two ways to proceed with this problem. The second and much easier way of doing this problem is to just deal with the stuff raised to the 4 th power and see what we get.

The substitution in this case would be. Two things to note here. A common mistake at this point is to lose it. Recognizing this can save a lot of time in working some of these problems. Note that the one third in front of the integral came about from the substitution on the differential and we just factored it out to the front of the integral. This is what we will usually do with these constants.

However, it looks like if we use the following substitution the first two issues are going to be taken care of for us. The most important thing to remember in substitution problems is that after the substitution all the original variables need to disappear from the integral. Note as well that this includes the variables in the differential!

This is a pretty good indication that we can use the denominator for our substitution so. Remember that we can just factor the 3 in the numerator out of the integral and that makes the integral a little clearer in this case.

The integral is very similar to the previous one with a couple of minor differences but notice that again if we differentiate the denominator we get something that is different from the numerator by only a multiplicative constant. Now, this one is almost identical to the previous part except we added a power onto the denominator.

Be careful in this case to not turn this into a logarithm. After working problems like the first two in this set a common error is to turn every rational expression into a logarithm. The idea that we used in the last three parts to determine the substitution is not a bad idea to remember.

Now, this part is completely different from the first three and yet seems similar to them as well.In calculusintegration by substitutionalso known as u -substitution or change of variables[1] is a method for evaluating integrals. Direct application of the fundamental theorem of calculus to find an antiderivative can be quite difficult, and integration by substitution can help simplify that task. It is the counterpart to the chain rule for differentiationin fact, it can loosely be thought of as using the chain rule "backwards.

This becomes especially handy when multiple substitutions are used. Before stating the result rigorously, let's examine a simple case using indefinite integrals.

This procedure is frequently used, but not all integrals are of a form that permits its use. In any event, the result should be verified by differentiating and comparing to the original integrand. For definite integrals, the limits of integration must also be adjusted, but the procedure is mostly the same. This equation may be put on a rigorous foundation by interpreting it as a statement about differential forms. One may view the method of integration by substitution as a partial justification of Leibniz's notation for integrals and derivatives.

The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be read from left to right or from right to left in order to simplify a given integral.

When used in the former manner, it is sometimes known as u -substitution or w -substitution in which a new variable is defined to be a function of the original variable found inside the composite function multiplied by the derivative of the inner function. The latter manner is commonly used in trigonometric substitutionreplacing the original variable with a trigonometric function of a new variable and the original differential with the differential of the trigonometric function.

Integration by substitution can be derived from the fundamental theorem of calculus as follows.

Hence the integrals. Since f is continuous, it has an antiderivative F. Applying the fundamental theorem of calculus twice gives. Alternatively, one may fully evaluate the indefinite integral see below first then apply the boundary conditions.

We thus have. Substitution can be used to determine antiderivatives.

**Trigonometric substitution formulas 1 بالعربي**

When evaluating definite integrals by substitution, one may calculate the antiderivative fully first, then apply the boundary conditions. In that case, there is no need to transform the boundary terms.

The tangent function can be integrated using substitution by expressing it in terms of the sine and cosine:. One may also use substitution when integrating functions of several variables. Here the substitution function v 1This formula expresses the fact that the absolute value of the determinant of a matrix equals the volume of the parallelotope spanned by its columns or rows.

More precisely, the change of variables formula is stated in the next theorem:. The conditions on the theorem can be weakened in various ways. For Lebesgue measurable functions, the theorem can be stated in the following form: [6].

Another very general version in measure theory is the following: [7] Theorem. In geometric measure theoryintegration by substitution is used with Lipschitz functions. By Rademacher's theorem a bi-Lipschitz mapping is differentiable almost everywhere. The following result then holds:. The above theorem was first proposed by Euler when he developed the notion of double integrals in The result is. From Wikipedia, the free encyclopedia.

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### 4.1: Integration by Substitution

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