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

4.2.8 · Maths › Calculus II — Integration

YEH KAAM KARTA HI KYUN HAI?

Is sab ke neeche engine hai u-substitution. Substitution ke liye chahiye ki integrand kuch aisa dikhe: $\int f (u) \frac{d u}{d x} d x .$ Toh hume chahiye ek hissa jo "andar wali cheez" ho ( $u$ ) aur doosra hissa jo "uski derivative" ho ( $d u$ ).

Trig ke liye, do derivative facts jo sab kuch drive karte hain:

Aur do Pythagorean identities jo baaki ko convert karti hain:

CASE A — $\int sin^{m} x cos^{n} x d x$

Parity KYUN? Odd power ka matlab hai tum ek factor alag kar sako aur phir bhi ek even power bachegi, aur even powers hi woh cheez hain jo identities $sin^{2} / cos^{2}$ rewrite kar sakti hain (unhe sirf squares pata hain).

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

CASE B — $\int tan^{m} x sec^{n} x d x$

Yeh splits KYUN? Kyunki $d u (tan) = sec^{2} x d x$ ke liye ek spare $sec^{2}$ chahiye, aur $d u (sec) = sec x tan x d x$ ke liye ek spare $sec tan$ chahiye. Identities sirf even $tan$ ya even $sec$ replacements deti hain, isliye hum woh factor bachate hain jo even leftover chhodta hai.

Forecast-then-Verify drill

Common mistakes (Steel-manned)

Active-recall flashcards

\int sin^{m} cos^{n}

, cosine power odd — kaunsa

u

u = sin x

; ek

cos x

d u

ke liye bachao, baaki

cos^{2} = 1 - sin^{2}

se convert karo.

\int sin^{m} cos^{n}

, dono powers even — method?

Power reduction:

sin^{2} = \frac{1 - c o s 2 x}{2}, cos^{2} = \frac{1 + c o s 2 x}{2}

\int tan^{m} sec^{n}

, secant power even — kaunsa

u

u = tan x

;

sec^{2} x

bachao,

sec^{2} = 1 + tan^{2}

se convert karo.

\int tan^{m} sec^{n}

, tangent power odd — kaunsa

u

u = sec x

;

sec x tan x

bachao,

tan^{2} = sec^{2} - 1

se convert karo.

\int tan x d x = ?

ln ∣ sec x ∣ + C

(yaani

- ln ∣ cos x ∣ + C

\int sec x d x = ?

ln ∣ sec x + tan x ∣ + C

(

\frac{s e c x + t a n x}{s e c x + t a n x}

se multiply karo).

\int sin^{2} x d x = ?

\frac{x}{2} - \frac{s i n 2 x}{4} + C

\int tan^{2} x d x = ?

tan x - x + C

tan^{2} = sec^{2} - 1

use karke.

tan

sec

mein

sec x tan x

bachate waqt leftover rebuild karne wali identity kaunsi hai?

tan^{2} x = sec^{2} x - 1

Recall Feynman: 12-saal ke bacche ko samjhao

Socho integral ek Lego tower hai $sin$ aur $cos$ bricks ka. Use saaf tarike se todne ke liye, tum ek special brick nikaalte ho jo "machine" ( $d u$ ) se match kare. Agar tumhare paas odd number mein cosine bricks hain, ek cosine nikaalo — ab ek even pile bachi, aur humara ek jadui rule hai ( $cos^{2} = 1 - sin^{2}$ ) jo even cosine piles ko sine piles mein badal deta hai. Phir hum $sin x$ ko $u$ rename karte hain aur yeh seedha algebra ban jaata hai ( $u^{2} - u^{4}$ wagera). $tan$ / $sec$ ke liye bhi yahi trick — woh brick nikalo jo derivative ke barabar hai, baaki convert karo. Agar sab kuch even hai aur kuch bhi clean peel nahin hota, toh hum "power adha karo" wala rule (power reduction) use karte hain.

Connections

u-substitution — woh core mechanism jis par har case reduce hota hai.
Pythagorean identities — $sin^{2} + cos^{2} = 1$ aur $1 + tan^{2} = sec^{2}$ .
Power-reduction & double-angle formulas — both-even case handle karta hai.
Trigonometric substitution — ulta idea, $a^{2} - x^{2}$ type integrals ke liye use hota hai.
Reduction formulas — bade $sin^{n} / cos^{n}$ powers ke liye systematic recursion.
Integration by parts — $\int sec^{3} x d x$ ke liye alternative route.