Here you will learn what is integration by substitution method class 12 with examples. Let’s begin – Integration By Substitution The method of evaluating an integral by reducing it to standard form by a proper substitution is called integration by substitution. If \(\phi(x)\) is continuously differentiable function, then to evaluate integrals of the form \(\int\) …
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Walli’s Formula : If m,n \(\in\) N & m, n \(\ge\) 2, then (a) \(\int_{0}^{\pi/2}\) \(sin^nx\)dx = \(\int_{0}^{\pi/2}\) \(cos^nx\)dx = \((n-1)(n-3)….(1 or 2)\over {n(n-2)….(1 or 2)}\) K where K = \(\begin{cases} \pi/2 & \text{if n is even}\ \\ 1 & \text{if n is odd}\ \end{cases}\) (b) \(sin^nx.cos^mx\)dx = \([(n-1)(n-3)….(1 or 2)][(m-1)(m-3)….(1 or 2)]\over {(m+n)(m+n-2)(m+n-4)….(1 or …
Newton Leibnitz formula If h(x) and g(x) are differentiable functions of x then, \(d\over dx\) \(\int_{g(x)}^{h(x)}\) f(t)dt = f[h(x)].h'(x) – f[g(x)].g'(x) Example : Evaluate \(d\over dt\) \(\int_{t^2}^{t^3}\) \(1\over log x\) dx Solution : We have, \(d\over dt\) \(\int_{t^2}^{t^3}\) \(1\over log x\) dx = \(1\over log t^3\) \(d\over dt\) \((t^3)\) – \(1\over log t^2\) \(d\over dt\) …
Here you will learn proof of integration of sec inverse x and cosec inverse x. Let’s begin – Integration of Sec Inverse x The integration of sec inverse x or arcsec x is \(xsec^{-1}x\) – \(log|x + \sqrt{x^2 – 1}|\) + C Where C is the integration constant. i.e. \(\int\) \(sec^{-1}x\) = \(xsec^{-1}x\) – \(log|x …
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Here you will learn proof of integration of cot inverse x or arccot x and examples based on it. Let’s begin – Integration of Cot Inverse x The integration of cot inverse x or arccot x is \(xcot^{-1}x\) + \(1\over 2\) \(log |1 + x^2|\) + C Where C is the integration constant. i.e. \(\int\) …
Here you will learn proof of integration of tan inverse x or arctan x and examples based on it. Let’s begin – Integration of Tan Inverse x The integration of tan inverse x or arctan x is \(xtan^{-1}x\) – \(1\over 2\) \(log |1 + x^2|\) + C Where C is the integration constant. i.e. \(\int\) …
Here you will learn proof of integration of cos inverse x or arccos x and examples based on it. Let’s begin – Integration of Cos Inverse x The integration of cos inverse x or arccos x is \(xcos^{-1}x\) – \(\sqrt{1 – x^2}\) + C Where C is the integration constant. i.e. \(\int\) \(cos^{-1}x\) = \(xcos^{-1}x\) …
Here you will learn proof of integration of sin inverse x or arcsin x and examples based on it. Let’s begin – Integration of Sin Inverse x The integration of sin inverse x or arcsin x is \(xsin^{-1}x\) + \(\sqrt{1 – x^2}\) + C Where C is the integration constant. i.e. \(\int\) \(sin^{-1}x\) = \(xsin^{-1}x\) …
Here you will learn what is the integration of log x dx with respect to x and examples based on it. Let’s begin – Integration of Log x The integration of log x with respect to x is x(log x) – x + C. where C is the integration Constant. i.e. \(\int\) log x dx …
Here you will learn proof of integration of cosecx or cosec x and examples based on it. Let’s begin – Integration of Cosecx or Cosec x The integration of cosec x is log |cosec x – cot x| + C or \(log |tan {x\over 2}|\) + C. where C is the integration constant. i.e. \(\int\) …
