Integrating Complex Functions: Trapezoidal Rule Explained

[2345qwert]

"2 examples which you cannot integrate with your knowledge so far are ∫1/1+x2 and ∫√1+x3. You need to use the trapezoidal rule for this" This was written in my textbook and I really can't understand what it means! I mean why can't we integrate it in the normal basic way?

 

[SteamKing]

Well, which integration techniques have you studied? Have you studied integration by parts? Trigonometric substitutions?

 

[2345qwert]

No, i haven't studied integration by parts or trigonometry substitutions yet. I have studied integration of exponentials and logarithms and simple functions.

 

[SteamKing]

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]

"2 examples which you cannot integrate with your knowledge so far are ∫1/1+x2 and ∫√1+x3.

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 + x<sup>3/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.</sup>

 

[2345qwert]

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.

What are the integration techniques that I could use to find the antiderivatives of these functions, apart from the trapezoidal rule??

 

[2345qwert]

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 + x<sup>3/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.</sup>

^{Thanks a lot for all your advice! But why is dx really important and what's the reason that it's there?}

 

[SteamKing]

What are the integration techniques that I could use to find the antiderivatives of these functions, apart from the trapezoidal rule??

The trapezoidal rule does not explicitly find the antiderivative of a given function; it provides a numerical means of evaluating a given definite integral.

 

[SteamKing]

Thanks a lot for all your advice! But why is dx really important and what's the reason that it's there?

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]

What are the integration techniques that I could use to find the antiderivatives of these functions, apart from the trapezoidal rule??

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)

 

[2345qwert]

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.

Okay! Thanks.

 

[2345qwert]

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)

Wow! That is so complicated. But thanks for your help.