Anti Derivative Of Arctan Anti Derivative Of Arctan What is the method of finding Antiderivative of Arctan? It is very simple let’s start learning about the method of finding the Antiderivative of Arctan which can also be written as ∫ tan^-1 x. For finding ∫ tan^-1 x we will use derivative of trigonometric identities and the by parts method according to which ∫f(x) * g(x) = f(x) ∫ g(x) - ∫d/dx f(x)* ∫g(x) dx. It is very simple let’s startlearning about the method of finding the Antiderivative of Arctan which can also bewritten as ∫ tan^-1 x. For finding ∫ tan^-1 x we will use derivative of trigonometric identities and the by partsmethod according to which ∫f(x) * g(x) = f(x) ∫ g(x) - ∫d/dx f(x)* ∫g(x) dx. For using this method we have to first decide which function between f(x) and g(x) willbe considered as a first function and which will be second function. For choosing first function and second function between f(x) and g(x) we use a simpleand very useful abbreviated form known as ILATE whereI=inverse function (cos^-1, tan^-1 etc.)L= logarithmic function (log x )A = arithmetic function (x^2, x^3+8x etc.).
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T = trigonometric function (sin x, cos x)E = exponential function (e^x)We can write ∫ tan^-1 x as: Find the antiderivative for the given function f(x) = x4 +cot x?For solving Antiderivative we need to follow the steps shown below: Step 1: In the first step we write the given function.f(x) = x4 +cot x, Step 2: Now we integrate the both side of the function,∫f(x) dx = ∫ x4 +cot x dx, Step 3: In this step we will separate the integral function.∫(x4 +cot x) dx = ∫x4 dx + ∫cot x dx, Step 4: After above step we will integrate each function with respect to ‘x’.∫(x4 +cot x) dx = x5/5 + ln|sin x| +c [Here x4 integration is x5/5 and Integration of cot xis ln|sin x|](Where ‘c’ is integration constant), At last we get the antiderivative of given functionx5/5 + ln|sin x| +c.Solving Initial Value problems in AntiderivativesAntiderivative is the term used in the calculus mathematics and especially in the topicof the Differential Equations. The anti derivatives are the type of the integralequations in which we don’t have limits on the Integration symbol. It is the reverseprocess of the derivatives or we can say it as the process of reverse differentiat. In calculus, an antiderivative, primitive integral or indefinite integral[1] of a function f is a function F whose derivative is equal to f, i.e., F ′ = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative.
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Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. Example of Antiderivative of Arctan The function F(x) = x3/3 is an antiderivative of f(x) = x2. As the derivative of a constant is zero, x2 will have an infinite number of antiderivatives; such as (x3/3) + 0, (x3/3) + 7, (x3/3) − 42, (x3/3) + 293 etc. Thus, all the antiderivatives of x2 can be obtained by changing the value of C in F(x) = (x3/3) + C; where C is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other; each graph's location depending upon the value of C. In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on).
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