In this video, I showed how to compute an indefinite integral of the square-root of a quadratic
Пікірлер: 35
@holyshit9223 ай бұрын
It is possible to calculate it by parts with clever choice of integration constant u = sqrt(5-4x-x^2) , dv = dx du = -(2+x)/sqrt(5-4x-x^2)dx , v = (x+2)
@PrimeNewtons3 ай бұрын
You always have a great substitute. There's something I learned from you, I will be using soon. Thanks.
@hassanejturay29943 ай бұрын
@@PrimeNewtonshow can I reach you, I have something to discuss with you
@rithvikarun71123 ай бұрын
How would u integrate -(x+2)^2/sqrt(5-4x-x^2) dx Nvm got it as an answer with cos-1(x+2/3)
@holyshit9223 ай бұрын
@@rithvikarun7112 I would rewrite it as -(x+2)^2=9-(5-4x-x^2) and I would have situatioon I = (x+2)sqrt(5-4x-x^2)+9Int(1/sqrt(5-4x-x^2),x) - I
@kushagrasharma59833 ай бұрын
Sir could you teach how to solve cubic equations? (Not by hit and trial method) 🙏🙏
@rithvikarun71123 ай бұрын
Yes please
@abhishankpaul3 ай бұрын
Cardan's method is one way. That is taught in UG courses. There is a very nasty formula though for direct calculations, you won't get any formula over 4th order polynomial. Mathematics restricts you upto that only
@boguslawszostak17843 ай бұрын
@@abhishankpaulThe method is not as nasty as it seems at first glance, but the result is often nasty and rarely useful for substitution into other formulas. In the first step, you substitute x=y+z, multiply and choose z so that the coefficient next to y^2 is equal to 0. Then You move y^3 to one side and the rest to the other side, obtaining the equation to solve: y^3=py+q. The second step is to substitute y=u+v. But You don't substitute it into the equation, but use the formula for (u+v)^3, which can be written as (u+v)^3= 3uv*(u+v)+u^3+v^3, Now you insert y= u+v into this formula and obtain y^3=3uvy+(u^3+v^3). The third step is to choose u and v in such a way as to obtain from the general formula from step 2 the equation being solved, i.e., p=3uv, q=(u^3+v^3). Now we calculate v from the first equation v=p/(3u) and substitute it into the second one: q=(u^3+(p/3u)^3)=u^3+p^3/(27u^3). This is an equation with an unknown u^3, which can be denoted by t. q=t+p/(27*t). This is a quadratic equation, from which we only need one solution. The equation u^3=t has three solutions, each of which gives you one solution to your equation.
@user-wh2kn1bp9q3 ай бұрын
Very good explanation. Thank you very much 🙂
@stigastondogg7303 ай бұрын
Ace of Base 10 - “I saw the sine”
@bazboy243 ай бұрын
Amazing video
@surendrakverma5553 ай бұрын
Thanks Sir
@antonionavarro10003 ай бұрын
Para que la expresión sqrt{5 - 4x -x^2} tenga sentido, se debe cumplir que 5 - 4x -x^2 >= 0 5 - 4x -x^2 >= 0 factorizando - (x+5) (x-1) >= 0 (x+5) (x-1) = 0) and (x-1 =-5 and x= 0 y por tanto cos(theta) >= 0. En consecuencia su valor absoluto no provoca ningún cambio, | cos(theta) | = cos(theta) No es que en el cambio de variable x-2 = 3•sin(x) se exija que sin(x) sea positivo, sino que, de hecho, ya lo es, atendiendo a su dominio.
@cliffordabrahamonyedikachi81753 ай бұрын
Thank you sir. Long answer.
@nullplan013 ай бұрын
Actually, I just did multiple substitutions. From Cardano's method, I know that it is possible to perform a linear substitution in a polynomial to lose the second highest term. A bit of playing around shows that in this case, the substitution x = t - 2 manages that, replacing the radicand with 9-t². Well, that is almost what I want, but not quite, so the second substitution I did was t = 3s. Because then the radicand becomes 9-9s², so I can bracket out the 9 and take its root separately. It joins the three from the fact that dt = 3 ds, and becomes a 9 in front of the integral. So now I have I = 9 ∫√(1-s²) ds And now finally I can do the trigonometric substitution I was aiming for all this time: Let s = sin u, ds = cos u du I = 9 ∫√(1-sin² u) cos u du = 9 ∫ cos² u du = 9 ∫ 1/2 (1 + cos 2u) du = 9/2 (u + 1/2 sin 2u) = 9/2 (u + sin u cos u) If s = sin u, cos u = √(1-s²) I = 9/2 (arcsin s + s √(1-s²)) = 9/2 (arcsin t/3 + t/3 √(1 - t²/9)) = 9/2 (arcsin (x+2)/3 + (x+2)/3 √(1 - (x+2)²/9)) + C
@biswambarpanda44683 ай бұрын
Long live sir..
@Aaryamank132Ай бұрын
Thanks you sir 🙏🙏🙏. You are a great teacher. I am living in india .
@happyhippo46642 ай бұрын
I like how the parenthesis after 9/2 mysteriously appeared. The chalk board has autocorrect.😀
@vinayakmamtani3 ай бұрын
We can just memorise the result for √a²-x²=x/2√a²-x² +a²/2sin-(x/a)
@Mephisto7073 ай бұрын
Theta can’t be on the 1st quadrant only, because you will not hit every x in the domain of the function. Theta must be in the 1st AND 4th quarters. It just so happens cosine is positive on those 2 quadrants.
@PrimeNewtons3 ай бұрын
Correct. Now I have to make a video explaining what you just said.
@abhishankpaul3 ай бұрын
@@PrimeNewtons don't delete this video if you make a new one. Provide link of this video over there and new video's link here
@jumpman82823 ай бұрын
@@PrimeNewtons When you do, keep in mind that 9 = (±3)², which means that (𝑥 + 2)² ∕ 9 = (±(𝑥 + 2) ∕ 3)². Also, −sin 𝜃 = sin(−𝜃). Thereby, with sin 𝜃 = (𝑥 + 2) ∕ 3, we can let 𝜃 ∈ Q1 because the negative sign will take care of the angles in Q4.
@levysarah29543 ай бұрын
On t aime Sir Newton.continue a nous régaler avec tes vidéos
@hassanejturay29943 ай бұрын
❤
@okmotivated47863 ай бұрын
Brother it's my humble request that please solve this question If the last three 3 digits of x⁴ is (x-58)², then what is the last digit of x is?
@korayoduncu58833 ай бұрын
The only thing i did not understand was how that cos square tetha turnes into (1/2 + 1/2 cos2theta)
@mil91023 ай бұрын
Trig identity
@rogerkearns80943 ай бұрын
mil9102's right and there are proofs in textbooks. We just learn the formula and use it when we need to. The one for sin squared is half minus half cos 2 theta.
@kevinmadden16453 ай бұрын
Cos(2 theta) equals 2(cos squared theta)-1. Solve for (cos squared theta) and the result is readily obtained.
@user-hl7lt8ge7o3 ай бұрын
(X+2)/3 is variety from -INF to +INF. Sin(Tetha) is variety from -1 to +1. Therefore I don’t understand how is possible to equivalent (x+2)/3=Sin(Tetha). Please discribe me it.
@user-oq7dp5sv8b3 ай бұрын
Can't we just use the standard integral II formula? It was possible after the very second step.