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 some integration examples for better understanding of integration concepts. Example 1 : Evaluate : \(\int\) \(dx\over {3sinx + 4cosx}\) Solution : I = \(\int\) \(dx\over {3sinx + 4cosx}\) = \(\int\) \(dx\over {3[{2tan{x\over 2}\over {1+tan^2{x\over 2}}}] + 4[{1-tan^2{x\over 2}\over {1+tan^2{x\over 2}}}]}\) = \(\int\) \(sec^2{x\over 2}dx\over {4+6tan{x\over 2}-4tan^2{x\over 2}}\) let \(tan{x\over 2}\) = …
Here, you will learn formulas for definite integrals and properties of definite integrals with examples. Let’s begin – A definite integral is denoted by \(\int_{a}^{b}\) f(x)dx which represent the algebraic area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis. Properties and Formulas for Definite Integrals …
