15 Integration
📐 समाकलन के सभी सूत्र (Integration Formulas)
✓ प्रत्येक सूत्र के साथ दो-दो उदाहरण | ✓ MathJax LaTeX संकेतन
🧮 1. मूलभूत समाकल सूत्र (Basic Integration)
\[\int 0 \ dx = C\]
उदा.1 \(\int 0 \,dx = C\)
उदा.2 \(\int 0 \cdot x^2 \,dx = C\)
\[\int 1 \ dx = x + C\]
उदा.1 \(\int 1 \,dx = x + C\)
उदा.2 \(\int 5 \,dx = 5x + C\)
\[\int x^n dx = \frac{x^{n+1}}{n+1}+C \quad (n \ne -1)\]
उदा.1 \(\int x^3 dx = \frac{x^4}{4}+C\)
उदा.2 \(\int x^{-2} dx = -\frac{1}{x}+C\)
\[\int \frac{1}{x} dx = \ln|x|+C\]
उदा.1 \(\int \frac{2}{x} dx = 2\ln|x|+C\)
उदा.2 \(\int \frac{1}{x+1} dx = \ln|x+1|+C\)
\[\int e^x dx = e^x + C\]
उदा.1 \(\int e^x dx = e^x + C\)
उदा.2 \(\int 5e^x dx = 5e^x + C\)
\[\int a^x dx = \frac{a^x}{\ln a}+C \quad (a>0,a\ne1)\]
उदा.1 \(\int 2^x dx = \frac{2^x}{\ln 2}+C\)
उदा.2 \(\int 10^x dx = \frac{10^x}{\ln 10}+C\)
📐 2. त्रिकोणमितीय समाकल (Trigonometric Integrals)
\[\int \sin x \,dx = -\cos x + C\]
उदा.1 \(\int \sin(2x)dx = -\frac{1}{2}\cos(2x)+C\)
उदा.2 \(\int 3\sin x\,dx = -3\cos x+C\)
\[\int \cos x \,dx = \sin x + C\]
उदा.1 \(\int \cos(3x)dx = \frac{1}{3}\sin(3x)+C\)
उदा.2 \(\int \cos(x/2)dx = 2\sin(x/2)+C\)
\[\int \tan x \,dx = \ln|\sec x|+C\]
उदा.1 \(\int \tan(2x)dx = \frac{1}{2}\ln|\sec(2x)|+C\)
उदा.2 \(\int x\tan(x^2)dx = \frac{1}{2}\ln|\sec(x^2)|+C\)
\[\int \cot x \,dx = \ln|\sin x|+C\]
उदा.1 \(\int \cot(3x)dx = \frac{1}{3}\ln|\sin(3x)|+C\)
उदा.2 \(\int \cot(x/2)dx = 2\ln|\sin(x/2)|+C\)
\[\int \sec x \,dx = \ln|\sec x + \tan x|+C\]
उदा.1 \(\int \sec(2x)dx = \frac{1}{2}\ln|\sec(2x)+\tan(2x)|+C\)
उदा.2 \(\int_0^{\pi/4}\sec x\,dx\) (निश्चित)
\[\int \csc x \,dx = -\ln|\csc x + \cot x|+C\]
उदा.1 \(\int \csc(x/2)dx = -2\ln|\csc(x/2)+\cot(x/2)|+C\)
उदा.2 \(\int \csc x\,dx = \ln|\csc x - \cot x|+C\)
\[\int \sec^2 x \,dx = \tan x + C\]
उदा.1 \(\int \sec^2(4x)dx = \frac{1}{4}\tan(4x)+C\)
उदा.2 \(\int 2\sec^2 x\,dx = 2\tan x+C\)
\[\int \csc^2 x \,dx = -\cot x + C\]
उदा.1 \(\int \csc^2(2x)dx = -\frac{1}{2}\cot(2x)+C\)
उदा.2 \(\int 3\csc^2 x\,dx = -3\cot x+C\)
\[\int \sec x \tan x \,dx = \sec x + C\]
उदा.1 \(\int \sec(2x)\tan(2x)dx = \frac{1}{2}\sec(2x)+C\)
उदा.2 \(\int \sec x \tan x\,dx = \sec x+C\)
\[\int \csc x \cot x \,dx = -\csc x + C\]
उदा.1 \(\int \csc(3x)\cot(3x)dx = -\frac{1}{3}\csc(3x)+C\)
उदा.2 \(\int 2\csc x\cot x\,dx = -2\csc x+C\)
🔄 3. प्रतिलोम त्रिकोणमितीय समाकल (Inverse Trig.)
\[\int \frac{1}{\sqrt{1-x^2}}dx = \sin^{-1}x + C\]
उदा.1 \(\int \frac{1}{\sqrt{4-4x^2}}dx = \frac{1}{2}\sin^{-1}x+C\)
उदा.2 \(\int \frac{1}{\sqrt{1-9x^2}}dx = \frac{1}{3}\sin^{-1}(3x)+C\)
\[\int -\frac{1}{\sqrt{1-x^2}}dx = \cos^{-1}x + C\]
उदा.1 \(\int -\frac{1}{\sqrt{1-x^2}}dx = \cos^{-1}x+C\)
उदा.2 \(\int -\frac{2}{\sqrt{1-4x^2}}dx = \cos^{-1}(2x)+C\)
\[\int \frac{1}{1+x^2}dx = \tan^{-1}x + C\]
उदा.1 \(\int \frac{1}{4+4x^2}dx = \frac{1}{4}\tan^{-1}x+C\)
उदा.2 \(\int \frac{1}{1+9x^2}dx = \frac{1}{3}\tan^{-1}(3x)+C\)
\[\int -\frac{1}{1+x^2}dx = \cot^{-1}x + C\]
उदा.1 \(\int -\frac{1}{1+x^2}dx = \cot^{-1}x+C\)
उदा.2 \(\int -\frac{2}{1+4x^2}dx = \cot^{-1}(2x)+C\)
\[\int \frac{1}{|x|\sqrt{x^2-1}}dx = \sec^{-1}|x|+C\]
उदा.1 \(\int \frac{1}{|x|\sqrt{x^2-1}}dx = \sec^{-1}|x|+C\)
उदा.2 \(\int \frac{1}{|2x|\sqrt{4x^2-1}}dx = \frac{1}{2}\sec^{-1}|2x|+C\)
📊 4. लॉगरिदमिक समाकल (Logarithmic Integrals)
\[\int \ln x \,dx = x\ln x - x + C\]
उदा.1 \(\int \ln(2x)dx = x\ln(2x)-x+C\)
उदा.2 \(\int \ln(x^2)dx = 2x\ln x - 2x+C\)
\[\int \log_a x \,dx = x\log_a x - \frac{x}{\ln a}+C\]
उदा.1 \(\int \log_2 x\,dx = x\log_2 x - \frac{x}{\ln 2}+C\)
उदा.2 \(\int \log_{10}(3x)dx = x\log_{10}(3x) - \frac{x}{\ln10}+C\)
\[\int (\ln x)^n dx = x(\ln x)^n - n\int (\ln x)^{n-1}dx\]
उदा.1 \(\int (\ln x)^2 dx = x(\ln x)^2 - 2x\ln x+2x+C\)
उदा.2 \(\int (\ln x)^3 dx = x(\ln x)^3 - 3\int (\ln x)^2 dx\)
\[\int \frac{1}{x\ln x}dx = \ln|\ln x|+C\]
उदा.1 \(\int \frac{1}{2x\ln x}dx = \frac{1}{2}\ln|\ln x|+C\)
उदा.2 \(\int \frac{1}{x\ln(3x)}dx = \ln|\ln(3x)|+C\)
\[\int x\ln x \,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4}+C\]
उदा.1 \(\int 3x\ln x\,dx = \frac{3x^2}{2}\ln x - \frac{3x^2}{4}+C\)
उदा.2 \(\int x\ln(2x)dx = \frac{x^2}{2}\ln(2x) - \frac{x^2}{4}+C\)
\[\int \frac{\ln x}{x}dx = \frac{1}{2}(\ln x)^2 + C\]
उदा.1 \(\int \frac{\ln(2x)}{x}dx = \frac{1}{2}(\ln(2x))^2+C\)
उदा.2 \(\int \frac{\ln(x^2)}{x}dx = (\ln x)^2+C\)
⚡ 5. चरघातांकी समाकल (Exponential Integrals)
\[\int x e^x dx = e^x(x-1)+C\]
उदा.1 \(\int 2x e^x dx = 2e^x(x-1)+C\)
उदा.2 \(\int x e^{2x}dx = \frac{e^{2x}}{4}(2x-1)+C\)
\[\int e^{ax}\sin(bx)dx = \frac{e^{ax}(a\sin bx - b\cos bx)}{a^2+b^2}+C\]
उदा.1 \(\int e^x\sin x\,dx = \frac{e^x}{2}(\sin x-\cos x)+C\)
उदा.2 \(\int e^{2x}\sin(3x)dx = \frac{e^{2x}(2\sin3x-3\cos3x)}{13}+C\)
\[\int e^{ax}\cos(bx)dx = \frac{e^{ax}(a\cos bx + b\sin bx)}{a^2+b^2}+C\]
उदा.1 \(\int e^{x}\cos x\,dx = \frac{e^{x}}{2}(\cos x+\sin x)+C\)
उदा.2 \(\int e^{-x}\cos(2x)dx = \frac{e^{-x}(-\cos2x+2\sin2x)}{5}+C\)
🧬 6. विशेष बीजीय समाकल (Special Algebraic)
\[\int \frac{1}{x^2+a^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a}+C\]
उदा.1 \(\int \frac{1}{x^2+4}dx = \frac{1}{2}\tan^{-1}\frac{x}{2}+C\)
उदा.2 \(\int \frac{1}{x^2+1}dx = \tan^{-1}x+C\)
\[\int \frac{1}{x^2-a^2}dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C\]
उदा.1 \(\int \frac{1}{x^2-4}dx = \frac{1}{4}\ln\left|\frac{x-2}{x+2}\right|+C\)
उदा.2 \(\int \frac{1}{x^2-1}dx = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right|+C\)
\[\int \frac{1}{\sqrt{a^2-x^2}}dx = \sin^{-1}\frac{x}{a}+C\]
उदा.1 \(\int \frac{1}{\sqrt{9-x^2}}dx = \sin^{-1}\frac{x}{3}+C\)
उदा.2 \(\int \frac{1}{\sqrt{1-4x^2}}dx = \frac{1}{2}\sin^{-1}(2x)+C\)
\[\int \frac{1}{\sqrt{x^2+a^2}}dx = \ln\left|x+\sqrt{x^2+a^2}\right|+C\]
उदा.1 \(\int \frac{1}{\sqrt{x^2+4}}dx = \ln\left|x+\sqrt{x^2+4}\right|+C\)
उदा.2 \(\int \frac{1}{\sqrt{x^2+1}}dx = \ln\left|x+\sqrt{x^2+1}\right|+C\)
\[\int \frac{1}{\sqrt{x^2-a^2}}dx = \ln\left|x+\sqrt{x^2-a^2}\right|+C\]
उदा.1 \(\int \frac{1}{\sqrt{x^2-4}}dx = \ln\left|x+\sqrt{x^2-4}\right|+C\)
उदा.2 \(\int \frac{1}{\sqrt{x^2-1}}dx = \ln\left|x+\sqrt{x^2-1}\right|+C\)
🧩 7. खंडशः समाकलन एवं प्रतिस्थापन (By Parts & Substitution)
\[\int u \, dv = uv - \int v \, du\]
उदा.1 \(\int x\sin x\,dx = -x\cos x + \sin x + C\)
उदा.2 \(\int x e^x dx = x e^x - e^x + C\)
\[\int f(g(x))\,g'(x)dx = \int f(u)du,\quad u=g(x)\]
उदा.1 \(\int 2x e^{x^2}dx = e^{x^2}+C\) ( \(u=x^2\) )
उदा.2 \(\int \sin x \cos x\,dx = \frac{1}{2}\sin^2 x + C\) ( \(u=\sin x\) )
टिप्पणियाँ
एक टिप्पणी भेजें