Anti Derivative Of Arctan

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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 wil 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 wil 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 wil separate the integral function.(x4 +cot x) dx = x4 dx + cot x dx,
Step 4: After above step we wil 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 cal ed 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 wil have an infinite number of antiderivatives; such as (x3/3) + 0, (x3/3) + 7, (x3/3) - 42,
(x3/3) + 293 etc.
Thus, al 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.
Essential y, 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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