Trigonometric integral Puns

I oftentimes feel like there are fast methods to finding an answer but struggle in doing so. Odd exponents, even exponents, where do I go to do things in the most efficient manner? Any tricks?

So I have gotten pretty decently far in a problem, integral of sqrt (1-x^2)dx.

I have gotten to the point where I got sin(2sinx)/4 + sin(x)/2

My issue is that I can't figure out how to reestablish sqrt (1-x^2) back into the problem. Like wouldn't it just be resubstituting the sin(2sin(x)) for the original function, or am I missing something?

CMRB is the MRB constant   https://mathworld.wolfram.com/MRBConstant.html

The following is worked at https://www.wolframcloud.com/obj/bmmmburns/Published/twitter10232021.nb

MRB Trig Integrals

My nspire cx won't solve intergal of 0 to pi of sin(x). I know the answer is 2, but the calculator returns something like .086. It can solve other integrals fine (can solve integral of 3 to 10 of 5x^2); it just won't solve the trigonometric integral. How do I fix it?

My calculator is in radian mode. I don't think it's CAS either. It worked in the instructional video by TI (https://www.youtube.com/watch?v=vpZl0BkA2Jg) but even though I typed the exact same thing, it didn't work for me.

If anyone could help with this I'd really appreciate it. I can't seem to find any resources as to how to answer these kind of integrals. I know what the identities are, I just can't seem to put them together.

https://preview.redd.it/cqy9x4qy0ah61.png?width=1024&format=png&auto=webp&s=96a0a1e3bb92fc1524f9ca34eb905cfc266e8606

Given this problem:

https://preview.redd.it/yrf8471o4uh61.png?width=114&format=png&auto=webp&s=6fd39af61f3c7acdc048eff4a2303d25a96fce97

This is my solution

https://preview.redd.it/v5tiupbw5uh61.png?width=3046&format=png&auto=webp&s=2125be6eb34e09ab848efb59bdf0517aa854d7d0

This is the final answer of Symbolab which differs only with signs when compared with my answers.

https://preview.redd.it/goa3djee4uh61.png?width=597&format=png&auto=webp&s=a0298fbe0e499382d81b49a0ec69aff415bd2949

I reviewed my answer multiple times but the solution seems to be correct. I did a brute force of double-checking my solution using Symbolab and Mathway. I do not understand why the explicit/direct inputting in the Symbolab differs from that of my answer. Could you please explain why is this so? I believe I may have a knowledge gap that my answer did not meet that of the Symbolab. Considering that, we only differ in rewriting the trig identities but it seems not a coincidence to have that similar numbers but different signs.

Introduction

The program ESGRAPH plots the parametric equations containing integrals:

x(t) = ∫(cos f(x) dx, x = 0 to t)

y(t) = ∫(sin f(x) dx, x = 0 to t)

The program uses f(x) = a*x^p

When p = 2, the curve is a Euler Spiral (see source).

Casio fx-9750GIII Program: ESGRAPH

' 2020-12-22 EWS

Rad

' L1, X L2, Y

"F = A×X^P"

"FACTOR"?→A

"P=2, EULER SPIRAL"

"POWER"?→P

∫(A×cos(X^P),0,-2π)→X

∫(A×sin(X^P),0,-2π)→Y

{X}→List 1

{Y}→List 2

Menu "π÷R?","8",A,"16",B,"32",C

Lbl A: 8 → R: Goto 0

Lbl B: 16 → R: Goto 0

Lbl C: 32 → R : Goto 0

Lbl 0

For -2π+π÷R→T To 2π Step π÷R

∫(A×cos(X^P),0,-2π)→X

∫(A×sin(X^P),0,-2π)→Y

Augment(List 1, {X})→List 1

Augment(List 2, {Y})→List 2

(T+2π)÷(4π)×100→G

RndFix(G,0)→G

Locate 1,7,"PROGRESS: %"

Locate 11,7,G

Next

S-Gph1 DrawOn, xyLine, List 1, List 2, 1, Dot

DrawStat

Havil, Julian. Curves for the Mathematical Curious: An Anthology of the Unpredictable, Historical, Beautiful, and Romantic Princeton Universal Press. Princeton and Oxford. Princeton, NJ 2019 ISBN 978-0-691-18005-2

For examples and more details, visit my blog at Eddie's Math and Calculator blog: http://edspi31415.blogspot.com/2021/01/casio-fx-9750giii-trigonometric.html

Hello all! As I was working on linear algebra, I recalled the substitution theorem for integrals. My current intuition is in terms of differential forms, but I thought it important to have multiple ways to look at things, so I am currently looking over the substitution theorem for integrals. At the moment, I completely understand it for basic cases like ∫ (x^2+3)^4 * 2x dx , we see that the derivative of x^2 + 3 is in the integrand, and we see that if we say that u = x^2 + 3, and get rid of 2x, then F'(u) = f(u) du/dx, which is the same as our original integrand. Essentially, ∫ f(g(x))g'(x) dx = ∫ f(u) du where u = g(x). My question is what this looks like in a trigonometric substitution problem, such as example 1 in Paul's Notes on trig substitution. To be clear, *I am not asking how to do trigonometric substitution*, I am simply asking if someone could be so kind as to write out the ∫ f(g(x))g'(x) dx -> ∫ f(u) du here and provide a explanation on why this theorem works (without infinitesimals please).

If anyone could be so kind as to help me with this, I would greatly appreciate it! Thank you so much.

So I am taking a calculus course on coursera and I am struggling with this integral. I've spend a lot of time trying to find a mistake, maybe someone here can help me with it.

https://preview.redd.it/kz65hdbg4yf41.png?width=1068&format=png&auto=webp&s=141629911a9e6ef654ed8636d4d1a8b072a65b95

https://preview.redd.it/3me6bkjh4yf41.png?width=672&format=png&auto=webp&s=76c04d95ced295c3402c8b51deea61e99151ef6a

So I have a homework problem and don’t understand something very well. How would I integrate something like x(t) = sin(xt)cos^2(xt)?

To clarify, I’m not looking for the answer, but: how do I simplify this in order to integrate? How does the coefficient x affect the simplification and integration process?

The integral is x * sqrt(4+x) dx. It's easiest done using u-sub, but the problem is asking to find a way using trig sub. How?

Hey! I'm trying to solve an integral with an awkward expression and I couldn't come up with anything smarter than just expanding the square and routinely integrating cos^(n)-type functions. Is there a way to do it faster? Anyway, I didn't even finished the aboventioned solution since I'm thinking it's a waste of time.