{"id":515,"date":"2025-01-27T13:42:57","date_gmt":"2025-01-27T07:57:57","guid":{"rendered":"https:\/\/pokharelsugam.com.np\/bioinformatics\/?p=515"},"modified":"2025-01-27T15:33:01","modified_gmt":"2025-01-27T09:48:01","slug":"integration","status":"publish","type":"post","link":"https:\/\/pokharelsugam.com.np\/bioinformatics\/integration\/","title":{"rendered":"Integration"},"content":{"rendered":"\n<script type=\"text\/javascript\" async \n  src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\">\n<\/script>\n\n\n\n<p class=\"has-medium-font-size\">Constant Rule:<br>\\( \\int k \\, dx = k x + C \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Constant Multiple Rule:<br>\\( \\int k \\cdot f(x) \\, dx = k \\int f(x) \\, dx\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Power Rule:<br>\\( \\int x^n \\, dx = \\frac{x^{n+1}}{n+1} + C \\quad \\text{for } n \\neq -1 \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\"><br>\\( \\int (ax+b)^n \\, dx = \\frac{(ax+b)^{n+1}}{a(n+1)} + C \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Exponential Functions:<br>\\( \\int e^x \\, dx = e^x + C \\)<br>\\( \\int e^{ax} \\, dx = \\frac{e^{ax}}{a} + C \\)<br>\\( \\int xe^x \\, dx = e^x(x-1) + C \\)<br>\\( \\int a^x \\, dx = \\frac{a^x}{\\ln a} + C \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Logarithmic Function:<br>\\( \\int \\frac{1}{x} \\, dx = \\ln|x| + C \\)<br>\\( \\int \\ln x \\, dx = x(\\ln x-1) + C \\)<br>\\( \\int \\frac{1}{ax+b} \\, dx = \\frac{\\ln(ax+b)}{a} + C\\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Trigonometric Functions:<br>\\( \\int \\sin(x) \\, dx = -\\cos(x) + C \\)<br>\\( \\int \\cos(x) \\, dx = \\sin(x) + C \\)<br>\\( \\int \\sec^2(x) \\, dx = \\tan(x) + C \\)<br>\\( \\int \\csc^2(x) \\, dx = -\\cot(x) + C \\)<br>\\( \\int \\sec(x) \\cdot \\tan(x) \\, dx = \\sec(x) + C \\)<br>\\( \\int \\csc(x) \\cdot \\cot(x) \\, dx = -\\csc(x) + C \\)<br>\\( \\int \\tan(x) \\, dx = \\ln|\\sec(x)| + C \\)<br>\\( \\int \\sec(x) \\, dx = \\ln|\\sec(x) + \\tan(x)| + C \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">\\( \\int \\sin(ax+b) \\, dx = \\frac{-\\cos(ax+b)}{a} + C \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Sum\/Diff Rule:<br>\\( \\int \\left[ f(x) \\pm g(x) \\right] \\, dx = \\int f(x) \\, dx \\pm \\int g(x) \\, dx \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Product Rule:<br>\\( \\int \\left[ u \\cdot v \\right] \\, dx = u\\int v \\, dx &#8211; \\int[\\frac{du}{dx} \\times \\int v \\, dx] \\, dx  \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Integration by Parts:<br>\\( \\int u \\, dv = uv &#8211; \\int v \\, du \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Substitution Rule:<br>\\( \\int f(g(x)) g'(x) \\, dx = \\int f(u) \\, du \\quad \\text{where } u = g(x) \\)<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Inverse Trigonometric Functions:<br>\\( \\int \\frac{1}{\\sqrt{1-x^2}} \\, dx = \\arcsin(x) + C \\)<br>\\( \\int \\frac{-1}{\\sqrt{1-x^2}} \\, dx = \\arccos(x) 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[&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":null,"ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":null,"ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":null,"ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":null,"stick-header-meta":null,"header-above-stick-meta":null,"header-main-stick-meta":null,"header-below-stick-meta":null,"astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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