Improper integrals — Type I (infinite limits), Type II (discontinuous integrand)

4.2.11 · Maths › Calculus II — Integration

WHY iska zaroorat hai hume?

Type I — Infinite limits

HOW karte hain master result ${\int }_1^{\infty }\frac{1}{x^p}dx$ derive

Yeh ek result 80/20 powerhouse hai — yeh lagbhag har convergence test ke neeche hai.

Scratch se Derivation.

Step 1 — $\infty$ ko $t$ se replace karo. ${\int }_1^{\infty }\frac{dx}{x^p}={\lim }_{t\to \infty }{\int }_1^t{x}^{-p}dx.$ Yeh step kyun? Hum $\infty$ tak integrate karne se forbidden hain; $t$ ek real upper limit hai jise hum FTC se handle kar sakte hain.

Step 2 — antiderivative (case $p\ne 1$). Power rule: $\int x^{-p}dx=\frac{x^{-p+1}}{-p+1}=\frac{x^{1-p}}{1-p}$. ${\int }_1^t{x}^{-p}dx=\frac{t^{1-p}-1}{1-p}.$ Yeh step kyun? Finite interval $[1,t]$ pe FTC apply karo — ab bilkul legal hai.

Step 3 — limit lo. Jaise $t\to \infty$, $t^{1-p}$ ka behavior sab decide karta hai:

• Agar $p>1$: exponent $1-p<0$, toh $t^{1-p}\to 0$. Result $=\frac{0-1}{1-p}=\frac{1}{p-1}$ ✓ converges.
• Agar $p<1$: exponent $1-p>0$, toh $t^{1-p}\to \infty$ → diverges.

Step 4 — borderline $p=1$. Power rule fail hoti hai; $\int x^{-1}dx=\ln x$ use karo: ${\lim }_{t\to \infty }[\ln t-\ln 1]={\lim }_{t\to \infty }\ln t=\infty \Rightarrow \text{diverges}.$ Yeh kyun important hai: $1/x$ knife-edge hai — yeh bas thoda sa haarta hai. Isliye $p=1$ famous boundary hai.

Type II — Discontinuous integrand (blow-up)

Derivation (case $p\ne 1$). Blow-up left end $0$ pe hai: ${\int }_0^1{x}^{-p}dx={\lim }_{t\to 0^{+}}\frac{x^{1-p}}{1-p}{|}_t^1={\lim }_{t\to 0^{+}}\frac{1-t^{1-p}}{1-p}.$ Yeh step kyun? Spike $0$ pe hai, toh hum $t>0$ se dheere-dheere chhete hain.

• $p<1$: $1-p>0$ → $t^{1-p}\to 0$ → answer $\frac{1}{1-p}$ converges.
• $p>1$: $1-p<0$ → $t^{1-p}\to \infty$ → diverges.
• $p=1$: ${\int }_0^1\frac{dx}{x}={\lim }_{t\to 0^{+}}(\ln 1-\ln t)=+\infty$ diverges.

Worked examples

Flashcards

Connections

• Fundamental Theorem of Calculus — woh legality limit jise hum $t\to$ bad point ke saath extend karte hain.
• Limits at Infinity — har Type I, ${\lim }_{t\to \infty }$ pe reduce hoti hai.
• Comparison Test for Integrals — directly $p$-integral pe built hai.
• p-series and Integral Test — $\sum 1/n^p$ same $p>1$ rule se converge karta hai.
• Laplace Transform & Gamma Function — convergent Type I integrals se define hote hain.
• Probability Density Functions — require karte hain ${\int }_{-\infty }^{\infty }f=1$ converge kare.

Concept Map