Foundations — Trigonometric integrals — sinᵐ·cosⁿ cases, tan and sec cases

4.2.8 · D1 · Maths › Calculus II — Integration › Trigonometric integrals — sinᵐ·cosⁿ cases, tan and sec cases

Iss move ko karne se pehle, tumhe har symbol pehchanna aana chahiye. Yeh page har ek cheez zero se build karta hai, uss order mein jisme woh ek doosre pe depend karte hain. Yahan kuch bhi assumed nahi hai. Throughout, "the parent topic" ka matlab hai woh main note jiske neeche yeh page hai — Trigonometric integrals — jo yahan build kiye gaye har symbol ko use karta hai.

1. Angles, aur $\sin$ / $\cos$ actually hote kya hain

Figure dekho. Point ka horizontal coordinate $\cos x$ hai (coral segment); vertical coordinate $\sin x$ hai (mint segment). Yahi hai in dono words ka poora meaning:

• ==$\cos x$== = circle pe tum "kitna right/left" ho.
• ==$\sin x$== = circle pe tum "kitna up/down" ho.

Kyunki point radius $1$ ke circle par baitha hai, chhote right triangle pe (horizontal leg $\cos x$, vertical leg $\sin x$, hypotenuse $1$) Pythagoras theorem yeh kehta hai:

${\cos }^{2}x+{\sin }^{2}x=1.$

Yeh hai pehli Pythagorean identity — aur tumne abhi dekha ki yeh sach kyun hai: yeh radius-1 triangle par Pythagoras hai. (Aur Pythagorean identities mein.)

2. Powers aur notation: ${\sin }^{2}x$, ${\sin }^{m}x$

• ${\sin }^{3}x=(\sin x)\cdot (\sin x)\cdot (\sin x)$.
• ${\sin }^{m}x{\cos }^{n}x$ mein letters ==$m$ aur $n$== sirf whole numbers ke placeholders hain — "sine ki koi power times cosine ki koi power". $m$ aur $n$ likhne se hum ek rule state kar sakte hain jo ${\sin }^2{\cos }^3$, ${\sin }^4{\cos }^4$, aur har doosra combination ek saath cover kare.

3. Even vs odd — woh word jis par poora topic tika hai

Ek calculus page ko isse kyun faraq padta hai? Kyunki ek hi fact ki wajah se:

4. Derivative — ek slope machine

Figure mein, lavender curve $\sin x$ hai. Har point pe coral tangent line ki steepness $\cos x$ ki value hai, jo mint curve ke roop mein plot ki gayi hai. Jahan $\sin x$ sabse tezi se chadh raha hai ($x=0$ par), $\cos x=1$; jahan $\sin x$ apni peak par flat hai, $\cos x=0$. Yahi picture hai pehle do anchors ke peeche:

$\cos$ par minus kyun? $x=0$ ke thoda baad, circle par point upar ja raha hai lekin baayein bhi — uska horizontal coordinate $\cos x$ ghaat raha hai, toh uska rate of change negative hai. Yahi sign ki wajah hai ki $u=\cos x$ substitutions baad mein $-du$ le aati hain.

Topic ko yeh kyun chahiye: substitution (§7) ek aisa chunk dhundta hai integrand mein jo doosre chunk ki derivative ho. Yeh facts exactly bataate hain ki $\sin$ aur $\cos$ ka "derivative-shape" kaisa dikhta hai, taaki tum jaano kaunsi brick peel karni hai. $\tan$ aur $\sec$ ke anchors ko pehle woh functions chahiye — woh §5 mein aate hain.

5. $\tan$ aur $\sec$ — circle ke do aur words

Ab kyunki $\tan$ aur $\sec$ exist karte hain, hum unki derivatives state kar sakte hain — aur, crucially, yeh bata sakte hain ki kyun:

Yeh shapes kyun? $\tan x=\frac{\sin x}{\cos x}$ par quotient rule apply karo: $\frac{d}{dx}\frac{\sin x}{\cos x}=\frac{\cos x\cdot \cos x-\sin x\cdot (-\sin x)}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}={\sec }^{2}x.$ Aakhiri step ne numerator mein pehli Pythagorean identity use ki. Isi tarah $\sec x=(\cos x{)}^{-1}$ differentiate hokar $\frac{\sin x}{{\cos }^{2}x}=\frac{1}{\cos x}\cdot \frac{\sin x}{\cos x}=\sec x\tan x$ deta hai. Toh yeh anchors derived hain, memorised nahi.

Identity ${\sin }^{2}x+{\cos }^{2}x=1$ ko ${\cos }^{2}x$ se divide karne par doosri Pythagorean identity milti hai. Hum divide sirf tabhi kar sakte hain jab ==$\cos x\ne 0$== (zero se kabhi divide nahi kar sakte); un angles par jahan $\cos x=0$, $\tan x$ aur $\sec x$ undefined hainویسے bhi, toh identity exactly usi domain par state ki gayi hai jahan uske symbols ka matlab banta hai:

$\frac{{\sin }^{2}x}{{\cos }^{2}x}+\frac{{\cos }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}⟹{\tan }^{2}x+1={\sec }^{2}x.$

Topic ko yeh kyun chahiye: yeh $\tan$/$\sec$ world ka "leftover rebuild karo" wala version hai — ${\cos }^2=1-{\sin }^2$ ka counterpart.

6. Integral sign $\int$ aur $dx$

Figure integration ko curve ke neeche area ke roop mein dikhata hai: ${\int }_a^{b}fdx$ woh shaded region hai. Lekin hamare topic ke liye useful reading "undo a derivative" wali hai.

7. $u$-substitution — engine

Neeche ka schematic is exact move ko ek example par trace karta hai — upar se neeche padho. Lavender box starting integral hai. Coral arrow derivative-shaped factor $\cos xdx$ ko peel karta hai aur usse $du$ rename karta hai; mint arrow leftover ${\cos }^{2}x$ ko pehli Pythagorean identity use karke $1-u^2$ ke roop mein rebuild karta hai. Dono streams butter box mein jaati hain: $u$ mein ek plain polynomial integral.

8. Absolute value $|\cdot |$ aur logarithm $\ln$

Isliye $\int \tan xdx=-\ln |\cos x|+C$ mein bars hain: $\cos x$ quadrants II aur III mein negative ho jaata hai, aur $|\cos x|$ log ko har jagah defined rakhta hai.

Prerequisite map

Equipment checklist

Connections

• Parent topic — yeh page seedha usi mein jaata hai.
• u-substitution — §7 mein assemble kiya gaya engine.
• Pythagorean identities — §1 aur §5 ki identities.
• Power-reduction & double-angle formulas — both-even backup tool.
• Trigonometric substitution — ulta idea.
• Reduction formulas — high powers ke liye recursion.
• Integration by parts — $\int {\sec }^{3}xdx$ ke liye alternate route.