All Integration Formulas in Hindi
UPI ID:- achalup41-1@oksbi
संयुक्त फलन का समाकलन (Integration of the composite function)
\(\int x^{n}dx=\frac{x^{n+1}}{n+1}+C, n\neq 1\)
\(\int \frac{1}{x}dx=logx+C\)
\(\int e^{x}dx=e^{x}+C \)
'C' समाकलन स्थिरांक है।
त्रिकोणमितीय फलन का समाकलन (Integration formula of the Trigonometric function)
\(\int sin x\hspace{2mm} dx=-cos x+C\)
\(\int cos x\hspace{2mm} dx=sin x+C\)
\(\int sec^{2}x\hspace{2mm} dx =tan x+C\)
\(\int cosec^{2}x\hspace{2mm} dx=-cotx+C\)
\(\int secx \hspace{1mm} tanx\hspace{2mm}dx = secx+C\)
\(\int cosecx \hspace{1mm} cotx\hspace{2mm}dx = -cosecx+C\)
\(\int tanx \hspace{2mm}dx = log(secx)+C\)
\(\int cotx \hspace{2mm}dx = log(sinx)+C\)
\(\int cosecx \hspace{2mm}dx = log(tan\frac{x}{2})+C\)
\(\int secx \hspace{2mm}dx = log\hspace{1mm}tan\left ( \frac{\pi }{4}+\frac{x}{2} \right )+C\)
'C' समाकलन स्थिरांक है।
प्रतिलोम त्रिकोणमितीय फलन का समाकलन (Integration into Inverse Trigonometric Functions)
\(\int \frac{1}{\sqrt{1-x^{2}}}\hspace{2mm}dx=sin^{-1}x+C\)
\(\int -\frac{1}{\sqrt{1-x^{2}}}\hspace{2mm}dx=cos^{-1}x+C\)
\(\int \frac{1}{1+x^{2}}\hspace{2mm}dx=tan^{-1}x+C\)
\(\int -\frac{1}{1+x^{2}}\hspace{2mm}dx=cot^{-1}x+C\)
\(\int \frac{1}{x\sqrt{x^{2}-1}}\hspace{2mm}dx=sec^{-1}x+C \)
\(\int -\frac{1}{x\sqrt{x^{2}-1}}\hspace{2mm}dx=-cosec^{-1}x+C\)
\(\int \frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}tan^{-1}\left ( \frac{x}{a} \right )+C\)
\(\int \frac{1}{\sqrt{a^{2}-x^{2}}}dx=sin^{-1}\left ( \frac{x}{a} \right )+C\)
\(\int \frac{1}{x \sqrt{x^{2}-a^{2}}}dx=\frac{1}{a} sec^{-1}\left ( \frac{x}{a} \right )+C \)
'C' समाकलन स्थिरांक है।
प्रतिस्थापन द्वारा समाकलन (Integration by substitution)
\(\int\frac{1}{ax+b}dx=\frac{1}{a}log(ax+b)+C\)
\(\int e^{ax+b}dx=\frac{1}{a}e^{ax+b}+C\)
\(\int e^{ax+b}dx=\frac{1}{a}\frac{e^{ax+b}}{log_ee}+C\)
\(\int sin(ax+b)dx=-\frac{1}{a}cos(ax+b)+C\)
\(\int cos(ax+b)dx= \frac{1}{a}sin(ax+b)+C\)
\(\int sec^{2}(ax+b)dx= \frac{1}{a}tan(ax+b)+C\)
\(\int cosec^{2}(ax+b)dx= -\frac{1}{a}cot(ax+b)+C\)
'C' समाकलन स्थिरांक है।
कुछ विशेष समाकलन (Few Special integrations)
\(\int\frac{1}{\sqrt{x^2+a^2}}dx =log(x+\sqrt{x^{2}+a^{2}})+C \)
\(\int\frac{1}{\sqrt{x^2+a^2}}dx =log(x+\sqrt{x^{2}-a^{2}})+C \)
\(\int a^{x}\hspace{2mm}dx=\frac{a^{x}}{loga}+C\)
'C' समाकलन स्थिरांक है।
दो फलनों का समाकलन या खंडशः समाकलन (Integration of two functions or Integration by parts)
\(\int e^{ax}cos \hspace{1mm}bx \hspace{2mm}dx= \frac{e^{ax}}{a^{2}+b^{2}}(a\hspace{1mm}cos\hspace{1mm}bx+b\hspace{1mm}sin\hspace{1mm}bx)+C\)
\(\int e^{ax}sin \hspace{1mm}bx \hspace{2mm}dx= \frac{e^{ax}}{a^{2}+b^{2}}(a\hspace{1mm}sin\hspace{1mm}bx-b\hspace{1mm}cos\hspace{1mm}bx)+C\)
\(\int\frac{1}{x^{2}-a^{2}}dx=\frac{1}{2a}log\frac{x-a}{x+a}+C, when\hspace{1mm} x> a \)
\(\int\frac{1}{a^{2}-x^{2}}dx=\frac{1}{2a}log\frac{a+x}{a-x}+C, when\hspace{1mm} x < a \)
'C' समाकलन स्थिरांक है।
निश्चित समाकलन के प्रगुण (Properties of Definite Integrals)
\(\int_{a}^{b}f(x)dx = \int_{a}^{b}f(t)dt+C\)
\(\int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx, when\hspace{1mm}a > b\)
\(\int_{0}^{a}f(x)dx = \int_{0}^{a}f(a-x)dx+C\)
'C' समाकलन स्थिरांक है।
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