You can use reverse rules to find antiderivatives. The easiest antiderivative rules are the ones that are the reverse of derivative rules you already know. These are automatic, one-step antiderivatives with the exception of the reverse power rule, which is only slightly harder. You know that the derivative of sin x is cos xso reversing online grocery delivery services tells you that an antiderivative of cos x is sin x.

What could be simpler? In symbols, you write. You can also use the slightly more difficult reverse power rule. By the power rule for differentiation, you know that. Use y — 5 x 4 for your ram promaster key fob not working. Recall that the power rule says to. Bring the power in front where it will multiply the rest of the derivative. To reverse this process, you reverse the order of the two steps and reverse the math within each step. The reverse power rule does not work for a power of negative one. The reverse power rule works for all powers including negative and decimal powers except for a power of negative one. Instead of using the reverse power rule, you should just memorize that the antiderivative of.

## List of Antiderivatives

Test your antiderivatives by differentiating them. If you get back to your original function, you know your antiderivative is correct. With the antiderivative you just found and using the fundamental theorem, you can determine the area under 20 x 3 between, say, 1 and Ryan has taught junior high and high school math since He lives in Evanston, Illinois.Joyful learning starts here!

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Follow Us. Get help here.The last topic that we discussed in the previous section was the harmonic series. In that discussion we stated that the harmonic series was a divergent series.

It is now time to prove that statement. From the section on Improper Integrals we know that this is. So, just how does that help us to prove that the harmonic series diverges? Well, recall that we can always estimate the area by breaking up the interval into segments and then sketching in rectangles and using the sum of the area all of the rectangles as an estimate of the actual area.

The image below shows the first few rectangles for this area. Now note a couple of things about this approximation. First, each of the rectangles overestimates the actual area and secondly the formula for the area is exactly the harmonic series! In other words, the harmonic series is in fact divergent. When discussing the Divergence Test we made the claim that. Again, from the Improper Integral section we know that.

### List of Antiderivatives

We will once again try to estimate the area under this curve. We will do this in an almost identical manner as the previous part with the exception that instead of using the left end points for the height of our rectangles we will use the right end points. Here is a sketch of this case. This time, unlike the first case, the area will be an underestimation of the actual area and the estimation is not quite the series that we are working with.

This means we can do the following. With the harmonic series this was all that we needed to say that the series was divergent. Because the terms are all positive we know that the partial sums must be an increasing sequence. In other words. Therefore, the partial sums form an increasing and hence monotonic sequence. In the second section on Sequences we gave a theorem that stated that a bounded and monotonic sequence was guaranteed to be convergent.

This means that the sequence of partial sums is a convergent sequence. So, who cares right? Well recall that this means that the series must then also be convergent!

So, once again we were able to relate a series to an improper integral that we could compute and the series and the integral had the same convergence. We went through a fair amount of work in both of these examples to determine the convergence of the two series. The ideas in these two examples can be summarized in the following test. A formal proof of this test can be found at the end of this section. There are a couple of things to note about the integral test.

First, the lower limit on the improper integral must be the same value that starts the series.The fundamental theorem of calculus links the relationship between differentiation and integration. We have seen from finding the area that the definite integral of a function can be interpreted as the area under the graph of a function. It justifies our procedure of evaluating an antiderivative at the upper and lower bounds of integration and taking the difference.

The first part of the theorem shows that indefinite integration can be reversed by differentiation. It also proves that for every continuous function, there is an antiderivative integral. Let F be any antiderivative, or indefinite integral, for f on [a,b].

If f is a continuous function and is defined by. Every continuous function f has an antiderivative, and there is one antiderivative F x given by a definite integral of f with a variable upper boundary. Varying the lower boundary of the integrand will produce other antiderivatives.

One of the first things to notice about the fundamental theorem of calculus is that the variable of differentiation appears as the upper limit of integration in the integral. This makes sense because if we are taking the derivative of the integrand with respect to x, it needs to be in either or both the limits of integration.

If we are integrating a function with respect to x, the function we end up with needs to be in terms of x. The second part of this theorem tells us how to evaluate a definite integral assuming that f has an indefinite integral:. This part of the theorem is one that we use in finding area. It allows us to compute the definite integral of a function by using one of infinitely many antiderivatives, or indefinite integrals.

It is important to note that if F x is an antiderivative of f x and the function is defined on some interval, then every other antiderivative G x of f x differs from F x by a constant:.

If we recall the general power rule for integration, taking the integral of a function will give us a constant. So there are infinitely many different antiderivatives for any given function. Here is a harder example using the chain rule. This is asking for the derivative of the integrand from the interval [2, sin x ]. In other words, we need to find the original function. From chain rule, we take the derivative of the outside function G times the derivative of the inside function sinx.

If f is a continuous function and is defined by then and therefore Every continuous function f has an antiderivative, and there is one antiderivative F x given by a definite integral of f with a variable upper boundary. The second part of this theorem tells us how to evaluate a definite integral assuming that f has an indefinite integral: where F is any antiderivative or indefinite integral of continuous function f on a closed interval [a,b].

Here is a simple example of the fundamental theorem of calculus: It is important to note that if F x is an antiderivative of f x and the function is defined on some interval, then every other antiderivative G x of f x differs from F x by a constant: If we recall the general power rule for integration, taking the integral of a function will give us a constant. Fundamental Theorem of Calculus Examples Let's do a couple of examples using of the theorem.

Taking the derivative with respect to x will leave out the constant. From the first part of the fundamental theorem of calculus, we Since sin x is in our interval, we let sin x take the place of x We take the derivative of both sides with respect to x.

We then have our original function. Sign up for free to access more calculus resources like. Wyzant Resources features blogs, videos, lessons, and more about calculus and over other subjects. Stop struggling and start learning today with thousands of free resources!

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Indefinite Integral Calculator Solve indefinite integrals step-by-step. Correct Answer :. Let's Try Again :. Try to further simplify. Advanced Math Solutions — Integral Calculator, advanced trigonometric functions.