Integrals
Integration differentiation ka ulta (inverse) process hai. Agar differentiation "rate" deta hai, to integration "total" wapas laata hai — aur graph ke neeche ka area nikaalne me kaam aata hai.
1. Integration as Anti-derivative
Agar \(\dfrac{d}{dx}F(x) = f(x)\), to \(\displaystyle\int f(x)\,dx = F(x) + C\), jahan \(C\) constant of integration hai.
\(C\) isliye aata hai kyunki kisi bhi constant ka derivative \(0\) hota hai — isliye answer me ek constant ki azaadi rehti hai. Isi liye ise indefinite integral kehte hain.
2. Standard Integrals
\[ \int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1), \qquad \int \frac{1}{x}\,dx = \ln|x| + C \]
\[ \int e^{x}\,dx = e^{x} + C, \qquad \int \sin x\,dx = -\cos x + C, \qquad \int \cos x\,dx = \sin x + C \]
\[ \int \sec^{2}x\,dx = \tan x + C, \qquad \int \frac{1}{1 + x^{2}}\,dx = \tan^{-1}x + C \]
\[ \int (3x^{2} + 2x + 1)\,dx = x^{3} + x^{2} + x + C. \]
3. Method 1: Substitution
Jab integrand me koi function aur uska derivative dono dikhein, to substitution use karo. \(u = g(x)\) rakho, \(du = g'(x)\,dx\).
\(\displaystyle\int 2x\,(x^{2} + 1)^{3}\,dx\). Lo \(u = x^{2} + 1\), \(du = 2x\,dx\): \[ = \int u^{3}\,du = \frac{u^{4}}{4} + C = \frac{(x^{2}+1)^{4}}{4} + C. \]
4. Method 2: Integration by Parts
Do functions ke product ke liye: \[ \int u\,v\,dx = u\int v\,dx - \int\!\left(\frac{du}{dx}\int v\,dx\right)dx. \]
\(u\) chunne ke liye ILATE order yaad rakho: Inverse, Log, Algebraic, Trig, Exponential.
\(\displaystyle\int x\,e^{x}\,dx\). Lo \(u = x\), \(v = e^{x}\): \[ = x e^{x} - \int e^{x}\,dx = x e^{x} - e^{x} + C = e^{x}(x - 1) + C. \]
5. Method 3: Partial Fractions
Rational functions \(\dfrac{p(x)}{q(x)}\) ko chhote simple fractions me tod kar integrate karte hain. Jaise:
\[ \frac{1}{(x-1)(x+1)} = \frac{1}{2}\!\left(\frac{1}{x-1} - \frac{1}{x+1}\right). \]
6. Definite Integral
Fundamental Theorem of Calculus: agar \(\int f(x)\,dx = F(x)\), to \[ \int_{a}^{b} f(x)\,dx = F(b) - F(a). \]
Definite integral ek number deta hai (koi \(+C\) nahi) — aur \(a\) se \(b\) tak curve ke neeche ka area.
\[ \int_{0}^{2} x^{2}\,dx = \left[\frac{x^{3}}{3}\right]_{0}^{2} = \frac{8}{3} - 0 = \frac{8}{3}. \]
7. Useful Properties of Definite Integrals
\[ \int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx \]
\[ \int_{a}^{b} f(x)\,dx = \int_{a}^{b} f(a + b - x)\,dx \]
\[ \int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a - x)\,dx \]
Key Takeaways
- Integration = anti-derivative; indefinite integral me hamesha \(+C\) lagao.
- Teen main methods: substitution, by parts (ILATE), partial fractions.
- Definite integral \(\int_a^b f\,dx = F(b) - F(a)\) — ek number (area).
- Definite integral ki properties problems chhoti kar deti hain.