Basic integration rules — power, trig, exponential, log: Question bank

4.2.2 · D5 · Maths › Calculus II — Integration › Basic integration rules — power, trig, exponential, log

Upar ki figure $n = - 1$ par hole dikhati hai: power-rule denominator $n + 1$ zero se guzarta hai, toh formula exactly wahin blow up ho jaata hai jahan $ln ∣ x ∣$ ko over lena padta hai.

True or false — justify

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

har real number

n

ke liye hold karta hai.

False — ye

n = - 1

par fail karta hai, jahan

n + 1 = 0

aur tum zero se divide kar rahe hote; us single hole ko

\int x^{- 1} d x = ln ∣ x ∣ + C

fill karta hai.

Ek hi function ke do antiderivatives identical hone chahiye.

False — wo kisi bhi constant se differ kar sakte hain, kyunki differentiation constants ko

0

kar deta hai; jaise

x^{2}

aur

x^{2} + 7

dono ka derivative

2 x

hai.

\int sin x d x = cos x + C

False — ye

- cos x + C

hai; differentiate karke check karo:

\frac{d}{d x} (- cos x) = sin x

, jabki

\frac{d}{d x} cos x = - sin x

(galat sign).

\int e^{x} d x = e^{x} + C

kyunki

e^{x}

integration se unchanged rehta hai.

True —

e^{x}

differentiation ka fixed point hai (

\frac{d}{d x} e^{x} = e^{x}

), toh ulta padha jaaye toh ye integration se bhi unchanged hai.

\int a^{x} d x = a^{x} + C

kisi bhi base

a > 0

ke liye.

False — kyunki

\frac{d}{d x} a^{x} = a^{x} ln a

, toh woh answer

ln a

times zyada bada hai; sahi integral

\frac{a ^{x}}{l n a} + C

hai (sirf

a = e

ke liye ye unchanged rehta hai, kyunki

ln e = 1

Tum

+ C

kabhi nahi chod sakte, calculation ke beech mein bhi nahi.

False (overstated) — intermediate steps mein tum brevity ke liye

+ C

drop kar sakte ho, bas final answer mein restore karo; jo tumhe kabhi nahi karna hai woh ye hai ki ek finished indefinite integral bina

+ C

ke present karo, kyunki answer ek poori family of functions hai, sirf ek nahi.

\int \frac{1}{x} d x = ln x + C

fully correct answer hai.

False (incomplete) —

ln x

x < 0

ke liye undefined hai lekin

\frac{1}{x}

wahan exist karta hai; sahi answer

ln ∣ x ∣ + C

hai jo saare

x \neq = 0

ke liye valid hai.

Linearity se hum likh sakte hain

\int f (x) g (x) d x = (\int f d x) (\int g d x)

False — linearity derivative sum rule

(f + g)^{'} = f^{'} + g^{'}

se aati hai, jo cleanly reverse hoti hai; lekin derivative product rule

(f g)^{'} = f^{'} g + f g^{'}

mein ek extra cross-term hai, isliye ye nahi reverse hoti ek product of integrals mein — products ke liye Integration by parts ya Integration by substitution chahiye.

\int 5 d x = 5 x + C

True —

5

5 x^{0}

ki tarah treat karo; power rule deta hai

5 \cdot \frac{x ^{1}}{1} = 5 x

, ya bas note karo ki

\frac{d}{d x} (5 x) = 5

Spot the error

\int x^{- 3} d x = \frac{x ^{- 2}}{- 3} + C

Galat denominator — tum naye power

n + 1 = - 2

se divide karte ho, purane se nahi; answer hai

\frac{x ^{- 2}}{- 2} + C = - \frac{1}{2 x ^{2}} + C

\int sec^{2} x d x = sec x tan x + C

Ye ek integral ko ek derivative se confuse kar raha hai;

\frac{d}{d x} tan x = sec^{2} x

, toh

\int sec^{2} x d x = tan x + C

\int 2^{x} d x = \frac{2 ^{x + 1}}{x + 1} + C

Power rule ko galat tarike se exponential par apply kiya gaya — yahan

x

exponent hai, base nahi, toh power rule bilkul apply nahi hota; use karo

\int a^{x} d x = \frac{a ^{x}}{l n a} + C = \frac{2 ^{x}}{l n 2} + C

\int (3 x^{2}) (x) d x = (\int 3 x^{2} d x) (\int x d x) = x^{3} \cdot \frac{x ^{2}}{2}

Invalid — tum ek product ko term-by-term product ki tarah integrate nahi kar sakte; pehle multiply out karke

3 x^{3}

banaao, phir integrate karke

\frac{3 x ^{4}}{4} + C

paaoge.

\int cos x d x = - sin x + C

Sign error — minus sign sine integral ka hissa hai, cosine ka nahi;

\int cos x d x = sin x + C

kyunki

\frac{d}{d x} sin x = cos x

\int \frac{4}{x} d x = 4 \cdot \frac{x ^{0}}{0} + C

Zero se illegal division — ye exactly

n = - 1

wala case hai jahan power rule break karta hai;

4

bahar nikalo aur log rule use karo:

4 ln ∣ x ∣ + C

x \geq 0

ke liye,

\int x d x = \frac{2}{3} (x)^{3} + C

, aur ye bhi

= \frac{x ^{1/2 + 1}}{1/2} + C

Pehla correct hai — exponent ki tarah likhne par,

(x)^{3} = x^{3/2}

, toh ye

\frac{2}{3} x^{3/2} + C

ke barabar hai us domain

x \geq 0

par jahan

x

exist karta hai; lekin doosra purane power

\frac{1}{2}

se divide karta hai naaye

\frac{3}{2}

ki jagah, toh galat hai (sahi denominator

\frac{3}{2}

hai, deta hai

\frac{x ^{3/2}}{3/2} = \frac{2}{3} x^{3/2}

Why questions

Har indefinite integral par

+ C

kyun aana chahiye, lekin definite integral par nahi?

Indefinite integrals ek poori family name karte hain (khoya hua constant unknown hai); definite integral mein tum ek antiderivative

F

ko endpoints par evaluate karte ho, aur constant

F (b) - F (a)

mein cancel ho jaata hai, toh kabhi appear nahi hota — dekho Definite integrals & FTC.

Power rule "raise then divide" kyun karta hai, "lower then multiply" kyun nahi?

Kyunki differentiation power lower karta hai aur us se multiply karta hai; integration us ko reverse karta hai, toh hum power raise karte hain aur naye power se divide karte hain taaki woh factor cancel ho jaaye jo differentiation produce karta.

e^{x}

ko saare exponentials

a^{x}

mein se alag kyun maana jaata hai?

Kyunki

ln e = 1

, correction factor

\frac{1}{l n a}

1

ban jaata hai, toh

e^{x}

khud mein integrate hota hai; har doosre base ke saath ek

\frac{1}{l n a}

aata hai — yehi

e

ko "natural" base banata hai, dekho Natural log and exponential functions.

\int sin x d x

mein minus sign kyun aata hai lekin

\int cos x d x

mein nahi?

Derivatives ko ulta padho:

\frac{d}{d x} cos x = - sin x

, toh

sin x

paane ke liye tumhe

- cos x

chahiye; jabki

\frac{d}{d x} sin x = cos x

mein koi adjustment nahi chahiye.

ln ∣ x ∣

(absolute value ke saath) sirf

ln x

ki jagah kyun chahiye?

Kyunki

\frac{1}{x}

negative

x

ke liye bhi defined hai, lekin

ln x

nahi; absolute value antiderivative ko saare

x \neq = 0

tak extend karta hai aur

\frac{d}{d x} ln ∣ x ∣ = \frac{1}{x}

bhi rehta hai (upar ki two-branch picture dekho).

Integration mein product ya quotient rule kyun nahi hota?

Kyunki derivatives ka product rule,

(f g)^{'} = f^{'} g + f g^{'}

, mein do terms ka sum hota hai; ise reverse karna ek clean formula mein nahi ho sakta — ye Integration by parts ban jaata hai, aur chain-rule products ko reverse karna Integration by substitution ban jaata hai; dono linearity jaise tidy nahi hain.

Hum integral ko re-integrating ki jagah differentiate karke verify kyun karte hain?

Differentiation deterministic aur unique hai, toh ye ek clean check hai; agar

\frac{d}{d x} (answer)

integrand ke barabar aaye, toh answer unavoidable

+ C

tak correct hai.

Edge cases

\int x^{0} d x

kya hai, aur kya power rule apply hota hai?

Haan —

x^{0} = 1

, aur

n = 0 \neq = - 1

, toh

\frac{x ^{1}}{1} + C = x + C

; rule safe hai kyunki hum zero se divide nahi kar rahe.

Power rule exactly

n = - 1

par kya hota hai, aur kya replace karta hai use?

Denominator

n + 1

0

ban jaata hai, toh formula undefined ho jaata hai; nature us hole ko ek bilkul alag function

ln ∣ x ∣ + C

se patch karta hai (ye wahi blow-up hai jo doosri figure mein dikhaya gaya hai).

Kya

\int 1^{x} d x = \frac{1 ^{x}}{l n 1} + C

valid hai?

Nahi —

ln 1 = 0

, toh ye zero se divide karta hai; lekin

1^{x} = 1

saare

x

ke liye, toh bas constant integrate karo:

\int 1 d x = x + C

x = 0

par, kya

\int \frac{1}{x} d x = ln ∣ x ∣ + C

meaningful hai?

Nahi —

\frac{1}{x}

x = 0

par defined nahi hai, toh antiderivative sirf un intervals par valid hai jo

0

contain nahi karte; tum yahan origin ke "across" integrate nahi kar sakte.

Kya disconnected domains par do antiderivatives har piece par alag constants se differ kar sakte hain?

Haan — kyunki

\frac{1}{x}

do alag intervals (

x < 0

aur

x > 0

) par exist karta hai

0

par ek gap ke saath, true antiderivative

ln ∣ x ∣ + C_{1}

hai

x > 0

ke liye aur

ln ∣ x ∣ + C_{2}

hai

x < 0

ke liye, independent constants ke saath; ek single "

+ C

" likhna quietly assume karta hai ki dono branches par same shift hai, jo ek mild abuse hai (upar ki two-branch picture dekho).

\int 0 d x

kya hai?

C

— zero function kisi bhi constant ka derivative hai, toh iska antiderivative sirf ek arbitrary constant hai jisme koi

x

term nahi.

Connections

Basic integration rules — power, trig, exponential, log (woh rules jinhe ye traps stress-test karte hain)
Derivatives — basic rules (har trap differentiate-backwards karne se resolve hota hai)
Integration by substitution (composites ke liye "product rule" ki jagah kya aata hai)
Integration by parts (genuine products ke liye "product rule" ki jagah kya aata hai)
Definite integrals & FTC (jahan $+ C$ finally gayab ho jaata hai)
Natural log and exponential functions ( $ln ∣ x ∣$ aur $e^{x}$ subtleties ka source)
Trigonometric integrals (sign-sensitive trig cases ko extend karta hai)