Foundations — Integration by parts — derivation from product rule, LIATE mnemonic

4.2.7 · D1 · Maths › Calculus II — Integration › Integration by parts — derivation from product rule, LIATE m

Isse pehle ki aap ek integral ko doosre se trade kar sako, aapko exactly pata hona chahiye ki page par har ek ghungroo-sa symbol kya matlab rakhta hai. Yeh page har ek symbol ko kuch nahi se build karta hai, uss order mein jisme ek doosre par depend karte hain, taaki jab aap $\int udv=uv-\int vdu$ se milein toh koi bhi mark aapko surprise na kare.

0. Ek function $f(x)$ — starting object

Picture ek grid par curve ki hai: horizontal axis har possible input $x$ hai, aur har $x$ ke liye curve ki height output $f(x)$ hai.

1. Do functions ka product $uv$

Ek fixed $x$ par number line ki picture lo: $u$ ki height padho, $v$ ki height padho, aur un dono side-lengths se ek rectangle banao. Uss rectangle ka area hi uss point par $uv$ hai.

2. Derivative $\frac{d}{dx}$ aur $f^{\prime }(x)$ — steepness

Symbol $\frac{d}{dx}(\cdot )$ ek operator hai — ise padho "woh rate jis par $(\cdot )$ change hoti hai jab $x$ move karta hai." Picture ek tangent line hai jo curve ko touch karti hai; steep tangent matlab badi derivative, flat tangent matlab zero.

Rectangle figure (s02) ko phir se dekho. Use thoda sa widen karo: ek nayi patli strip upar dikhti hai ($u$ times $v$ mein change) aur ek side mein ($v$ times $u$ mein change). Yahi do strips precisely $u\frac{dv}{dx}$ aur $v\frac{du}{dx}$ hain.

3. Prime aur Leibniz notations $u^{\prime }$, $\frac{du}{dx}$, $du$

4. Integral $\int f(x)dx$ — accumulated area

Imagine karo ki curve ke neeche infinitely many patli rectangles ki height $f(x)$ aur width $dx$ ki stack kar rahe ho — unka total area integral hai.

5. Constant of integration $+C$

6. Fundamental Theorem of Calculus — hinge

7. $u$ aur $dv$ choose karna — woh letters jo aapko decide karwate hain

Parent note deciding order ke roop mein LIATE deta hai. Uss decision mein har symbol — $u$, $du$, $dv$, $v$ — upar zero se build ho chuka hai.

Foundations topic ko kaise feed karte hain

Equipment checklist

Khud ko test karo — right side cover karo aur zor se jawab do.

Connections

• Parent topic — Integration by parts
• Product rule (differentiation) — woh identity jis tak yahan har symbol le jaata hai.
• Fundamental Theorem of Calculus — woh hinge jo left side ko collapse karta hai.
• Integration by substitution — sibling technique jo same symbols par built hai.