You really need to get up to speed on how to write mathematical expressions.2345qwert said:"2 examples which you cannot integrate with your knowledge so far are ∫1/1+x2 and ∫√1+x3.
SteamKing said:Your text is assuming that you haven't studied the integration techniques which you would need to find the antiderivatives of those two particular functions.
Mark44 said:You really need to get up to speed on how to write mathematical expressions.
Exponents
I'm assuming that x2 and x3 mean x2 and x3, respectively. You can write exponents like these if you click the Go Advanced button just below the text entry pane. This opens the advanced menu across the top, which has an X2 button.
If this seems too complicated, you can use ^ to indicate a power such as x^2.
Parentheses
1/1 + x2 means (1/1) + x2, which is the same as 1 + x2. Since this is probably not what you meant, you need to add parentheses around the entire denominator, like this: 1/(1 + x2).
√1 + x3 means 1 + x3. To indicate that the square root includes the variable, use parentheses - √(1 + x3/SUP]).
Integrals
Neither of your integrals includes the differential, dx. At your stage of learning it might seem like a useless appendage, but trust me, it's there for a reason. If you get in the habit of omitting it, that practice will bite you in the butt later on.
2345qwert said:What are the integration techniques that I could use to find the antiderivatives of these functions, apart from the trapezoidal rule??
2345qwert said:Thanks a lot for all your advice! But why is dx really important and what's the reason that it's there?
2345qwert said:What are the integration techniques that I could use to find the antiderivatives of these functions, apart from the trapezoidal rule??
SteamKing said:It is important to include the 'dx' in an integral expression to indicate w.r.t. which variable the integration takes place.
In the expressions you have been exposed to so far, it's probably pretty obvious which is the integration variable. However, in a complicated integral expression with many different variables and other terms, it may not be obvious which variable you are integrating with respect to. It is good practice and a good habit to learn to express the dx or d(whatever) now early in your calculus instruction so that it becomes automatic later on.
AlephZero said:Wait till you learn how to differentiate log, trig, and exponential functions.
But there are many functions that look "simple" but do not have "simple" antiderivatives. That is the case for your example of ##\int \sqrt{1+x^3}\, dx##. Integrals like that can be expressed in terms of "elliptic functions", but you won't learn about them in the standard "calculus sequence" of courses.
(If you really want to know the answer, go to http://integrals.wolfram.com/index.jsp?expr=sqrt[1+x^3]&random=false)
The Trapezoidal Rule is a numerical method for approximating the value of a definite integral. It works by approximating the area under the curve of a function by dividing it into trapezoids and summing their areas. This method is commonly used when it is difficult or impossible to find an exact solution to a complex integral.
The Trapezoidal Rule differs from other numerical integration methods, such as Simpson's Rule or the Midpoint Rule, in that it uses straight lines to approximate the curve of the function. This makes it a simpler method to implement, but it may not be as accurate as other methods for certain types of functions.
One advantage of using the Trapezoidal Rule is that it is relatively easy to implement and does not require a lot of computational resources. It can also provide a good approximation for smooth functions with relatively few trapezoids, making it a quick and efficient method for obtaining an approximate solution to a complex integral.
Yes, there are some limitations to using the Trapezoidal Rule. It may not be as accurate as other numerical integration methods for certain types of functions, such as those with sharp corners or highly oscillatory behavior. Additionally, the accuracy of the approximation may decrease as the number of intervals used increases.
To improve the accuracy of the Trapezoidal Rule, you can use a smaller interval size, which means using more trapezoids to approximate the curve of the function. You can also use a more accurate method, such as Simpson's Rule or Gaussian Quadrature, if the function allows for it. Additionally, using a computer program or calculator to perform the calculations can reduce errors caused by human calculation.