{"id":22120,"date":"2024-01-10T16:36:31","date_gmt":"2024-01-10T11:06:31","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=22120"},"modified":"2024-01-11T12:35:55","modified_gmt":"2024-01-11T07:05:55","slug":"inter-2nd-year-maths-2b-integration-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2b-integration-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2B Integration Formulas"},"content":{"rendered":"

Use these Inter 2nd Year Maths 2B Formulas<\/a> PDF Chapter 6 Integration to solve questions creatively.<\/p>\n

Intermediate 2nd Year Maths 2B Integration Formulas<\/h2>\n

\u2192 Integration is the inverse process of differentiation.<\/p>\n

\u2192 Let A \u2286 R and let f: A \u2192 R be a function. If there is a function B on A such that F'(x) = f(x), \u2200 x \u2208 A, then we call B an antiderivative of for a primitive of f.
\ni.e., \\(\\frac{d}{d x}\\)(sin x) = cos x, \u2200 x \u2208 R ax
\nf(x) = cos x, x \u2208 R, then the function
\nF(x) = sin x, x \u2208 R is an antiderivative or primitive of f.<\/p>\n

\u2192 If F is an antiderivative of A, then for k \u2208 R, we have (F + k) (x) = f(x), \u2200 x \u2208 A.<\/p>\n

\u2192 Hence F + k is also an antiderivative off.
\n\u2234 c is any real number F(x + c) = G(x) = sin x + c, \u2200 x \u2208 R is also an antiderivative of cos x.<\/p>\n

\u2192 It is denoted by \u222b (cos x) dx = sin x + c, (i.e.) \u222b f(x) dx = F(x) + c.<\/p>\n

\"Inter<\/p>\n

\u2192 Here c is called a constant of integration,
\nf is called the integrand and x is called the variable of integration.<\/p>\n

Standard Forms:<\/strong><\/p>\n

\u2192 \u222bxn<\/sup> dx = \\(\\frac{x^{n+1}}{n+1}\\) + c if n \u2260 – 1<\/p>\n

\u2192 \u222b\\(\\frac{1}{x}\\) dx = log |x| + c<\/p>\n

\u2192 \u222b sin x dx = – cos x + c, x \u2208 R<\/p>\n

\u2192 \u222b cos x dx – sin x + c, x \u2208 R<\/p>\n

\u2192 \u222btan x dx = log |sec x| + c<\/p>\n

\u2192 \u222b cot x dx = log |sin x | + c<\/p>\n

\u2192 \u222bsec x dx = log |sec x + tan x | + c (or) log |tan \\(\\left(\\frac{\\pi}{4}+\\frac{x}{2}\\right)\\)| + c<\/p>\n

\u2192 \u222bcosec x dx = log |cosec x – cot x| + c (or) log |tan\\(\\left(\\frac{x}{2}\\right)\\)| + c<\/p>\n

\u2192 \u222bsec2<\/sup> x dx = tan x + c, x \u2208 R – \\(\\left|\\frac{n \\pi}{2}\\right|\\), n is odd integer<\/p>\n

\u2192 \u222bcosec2<\/sup> x dx = – cot x + c \u2192 R – n\u03c0, n \u2208 Z<\/p>\n

\u2192 \u222bsec x tan x dx = sec x + c, R – \\(\\left[\\frac{n \\pi}{2}\\right]\\), n is an odd integer<\/p>\n

\u2192 \u222bcosec x cot xdx = – cosec x + c, R – [n\u03c0], n \u2208 Z<\/p>\n

\u2192 \u222bex<\/sup> dx = ex<\/sup> + c, x \u2208 R<\/p>\n

\"Inter<\/p>\n

\u2192 \u222bax<\/sup> dx = \\(\\frac{a^{x}}{\\log _{e} a}\\) + c, a > 0, a \u2260 1<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{1+x^{2}}}\\) dx = sin-1<\/sup>x + C = – cos-1<\/sup> (x) + c<\/p>\n

\u2192 \u222b\\(\\frac{d x}{1+x^{2}}\\) dx = tan-1<\/sup>x + C = – cot-1<\/sup> (x) + c<\/p>\n

\u2192 \u222b\\(\\frac{d x}{|x| \\sqrt{x^{2}-1}}\\) dx = sec-1<\/sup>x + C = – cosec-1<\/sup> (x) + c<\/p>\n

\u2192 \u222b sinh x dx = cosh x + c<\/p>\n

\u2192 \u222bcosh xdx = sinh x + c<\/p>\n

\u2192 \u222bcosec2<\/sup>h x dx coth x + c<\/p>\n

\u2192 \u222bsec2<\/sup>h x dx = tanh x + c<\/p>\n

\u2192 \u222bcosech x coth xdx = – cosech x + c<\/p>\n

\u2192 \u222bsech x tanh x dx = – sech x + c<\/p>\n

\u2192 \u222beax<\/sup> dx = \\(\\frac{e^{a x}}{a}\\) + c<\/p>\n

\u2192 \u222beax+b<\/sup> dx = \\(\\frac{e^{a x+b}}{a}\\) + c<\/p>\n

\u2192 \u222bsin (ax + b) dx = \\frac{-\\cos (a x+b)}{a}\\(\\) + c<\/p>\n

\u2192 \u222bcos (ax + b) dx = \\(\\frac{\\sin (a x+b)}{a}\\) + c<\/p>\n

\u2192 \u222bsec2<\/sup> (ax + b) dx = \\(\\frac{\\tan (a x+b)}{a}\\) + c<\/p>\n

\u2192 \u222bcosec2<\/sup> (ax + b) dx = \\(\\frac{-\\cot (a x+b)}{a}\\) + c<\/p>\n

\u2192 \u222bcosec(ax + b) cot(ax + b) dx = \\(\\frac{-{cosec}(a x+b)}{a}\\) + c<\/p>\n

\u2192 \u222bsec (ax + b) tan(ax + b) dx = \\(\\frac{\\sec (a x+b)}{a}\\) + c<\/p>\n

\u2192 \u222bf(x).g(x) dx = f(x) \u222bg(x) dx – \u222b[\\(\\frac{d}{d x}\\) f(x) . \u222b g(x) dx] dx (called as integration by parts)<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{1+x^{2}}}\\) dx = sinh-1<\/sup> (x) + c, x \u2208 R = log (x + \\(\\sqrt{x^{2}+1}\\)) + c, x \u2208 R<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{x^{2}-1}}\\) dx = cosh-1<\/sup> (x) + c (or) log (x + \\(\\sqrt{x^{2}-1}\\)) + c, x \u2208 (1, \u221e)
\n= – cos h-1<\/sup> (- x) + c (or) log (x + \\(\\sqrt{x^{2}-1}\\)) + c, x \u2208 (1, \u221e)
\n= log |x + \\(\\sqrt{x^{2}-1}\\)| + c, x \u2208 R – [- 1, 1]<\/p>\n

\"Inter<\/p>\n

\u2192 \u222bex<\/sup> [f(x) + f'(x)] dx = ex<\/sup>. f(x) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{a^{2}-x^{2}}}\\) dx = sin-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{a^{2}+x^{2}}}\\) dx = sinh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c (or) log \\(\\frac{\\left|x+\\sqrt{x^{2}+a^{2}}\\right|}{a}\\) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{x^{2}-a^{2}}}\\) dx = cosh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c (or) log \\(\\frac{\\left|x+\\sqrt{x^{2}-a^{2}}\\right|}{a}\\) + c<\/p>\n

\u2192 \u222b\\(\\frac{1}{a^{2}+x^{2}}\\) dx = \\(\\frac{1}{a}\\) tan-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\)+ c<\/p>\n

\u2192 \u222b\\(\\frac{1}{\\sqrt{a^{2}-x^{2}}}\\) = \\(\\frac{1}{2a}\\) log \\(\\left|\\frac{a+x}{a-x}\\right|\\) + C<\/p>\n

\u2192 \u222b\\(\\frac{1}{x^{2}-a^{2}}\\) dx = \\(\\frac{1}{2a}\\) log \\(\\left|\\frac{x-a}{x+a}\\right|\\) + c<\/p>\n

\u2192 \u222b\\(\\sqrt{a^{2}-x^{2}}\\) dx = \\(\\frac{1}{2}\\)x \\(\\sqrt{a^{2}-x^{2}}\\) + \\(\\frac{a^{2}}{2}\\) sin-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \u222b\\(\\sqrt{x^{2}+a^{2}}\\) dx = \\(\\frac{1}{2}\\)x \\(\\sqrt{x^{2}+a^{2}}\\) + \\(\\frac{a^{2}}{2}\\) sinh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\u2192 \u222b\\(\\sqrt{x^{2}-a^{2}}\\) dx = \\(\\frac{1}{2}\\)x \\(\\sqrt{x^{2}-a^{2}}\\) + \\(\\frac{a^{2}}{2}\\) cosh-1<\/sup>\\(\\left(\\frac{x}{a}\\right)\\) + c<\/p>\n

\"Inter<\/p>\n

\u2192 To evaluate<\/p>\n