# Questions tagged [integration]

For questions related to the teaching of integral calculus

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### Why can an easily graphable definite integral, be labyrinthine to evaluate?

How can I explain to 16-year-olds, who just started calculus, why it's so nettlesome and tricky to symbolically integrate definite integrals, when their graphs look so unremarkable and straightforward?...
326 views

### Usefulness of $u$-substitution in and beyond early Calculus?

My students, when presented with an integral (source) like $$\int (2x+2)e^{x^2+2x+3} \ dx$$ are apt to recognize derivative patterns like $u' e^{u}$ and reverse-engineer anti-derivatives rather than ...
112 views

### Finding an error in a partial integration [closed]

There must be an error in this partial integration but I do not see it. Do you see it?
128 views

### Analogy for cylindrical shells

The analogy for cross-sections is easy since we can think of how slices of bread can make up a loaf. But what would be the analogy for cylindrical shells? Regarding shapes, apparently there's ...
112 views

### The purpose of a particular rational function integration exercise

This might be a more appropriate question for math.stackexchange, but it's about a problem I'm considering giving my students, so here it goes. One of the later exercises in Section 7.4 of James ...
543 views

### Intuition or geometry for Partial Fractions

When teaching partial fractions, there's probably no way to escape the heavy algebra necessary for partial fractions, but I'm wondering how to introduce the idea in a way that is intuitive or ...
365 views

### The hardest case of integration by partial fractions

The context is explaining to calculus students how to integrate rational functions by using partial fractions decomposition. As we all know, partial fractions decomposition is a method to write every ...
1k views

### A different symbol for the indefinite integral/antiderivative?

Examples. An indefinite integral (or antiderivative) of $\cos$ is $\sin$: $$\int \cos = \sin.$$ Edit: There has been much unexpected confusion with the above statement. I define the above statement ...
94 views

### Retain problems and combat regression in learning

Regressive Learning It's a really stressful situation. I can achieve but not retain expertise in maths problems. History 6 months back, I studied integration in Calculus at college. I learnt it all ...
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### Demonstrating that integrals of some unbounded functions exist, and others do not

This is my first year teaching calculus. On a recent quiz, I asked my students to give an argument that $\int^0_1(1/x)dx$ does not exist. I was looking for arguments that appealed to Riemann sum ...
2k views

### Should we teach trigonometric substitution?

This is the question that was not asked here. Also related is this question, but both presuppose that it will be taught and ask about how best to do it. My question here is, suppose we are designing ...
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### What is a good way to explain the Lebesgue integral to non-math majors?

A few days ago I had my last discussion session on probability theory as a TA. In the end I asked students to ask me questions as this is the last class. One of the student asked me about the (real) ...
126 views

### How useful/useless is the indefinite integral [duplicate]

After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community. Do you think it makes sense to teach indefinite integrals? My opinion is that ...