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Antiderivative Of Sinx

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					                         Antiderivative Of Sinx
Antiderivative Of Sinx

To calculate the Antiderivative of sinx, first of all we discuss definition of antiderivative.
Antiderivative is just opposite of derivative means if g(x) is a derivative of f(x), then
antiderivative of g(x) is f(x)-d (f(x)) = g(x) and antiderivative of g(x) means Integration of g(x) is
∫g(x) dx = f(x) + c,

dx For proving that antiderivative is an opposite operation of derivation, we take some
examples like we have a function f(x) = x. Step 1: Firstly we calculate derivative of f(x) is d (x)
= 1.x1-0 = 1 dx Step 2:

As we all know that antiderivative operation means integration operation, and if integration of
1 produces original function than we can say that antiderivative operation is opposite
operation of derivation. So, let’s check it: ∫1 dx = x0+1 + c = x + c

0+1 It produces x as a result, which is our original function f(x). So, we can say that
antiderivative is an opposite operation of derivation.

So, we can say that if somebody knows derivation very well, then it’s very easy for them to
calculate antiderivation operation on certain functions.
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The antiderivative is the name we sometimes, (rarely) give to the operation that goes
backward from the derivative of a function to the function itself.

Since the derivative does not determine the function completely (you can add any constant to
your function and the derivative will be the same), you have to add additional information to go
back to an explicit function as antiderivative.

Thus we sometimes say that the antiderivative of a function is a function plus an arbitrary
constant. Thus the antiderivative of cos x is (sin x) + c.

The more common name for the antiderivative is the indefinite integral. This is the identical
notion, merely a different name for it.

Find the antiderivative for the given function f(x) = x4 +sin x +1?

For solving Antiderivative we need to follow the steps shown below:
Step: In the first step we write the given function.
f(x)= x4 + sin x +1,

Step 2: Now we integrate the both side of the function
∫f(x) dx = ∫x4 + sin x +1 dx,

Step 3: In this step we separate the integral function.
∫(x4 + sin x +1) dx = ∫x4 dx + ∫sin x dx + ∫1 dx,

Step 4: After above steps we will integrate each function with respect to ‘x’.
∫(x4 + sin x +1) dx = x5/5 – cos x + x +c [Here x4 integration is x5/5, and Integration of sin x is
–cos x and integration of 1 is ‘x’].
(Where ‘c’ is integration constant)
At last we get the antiderivative of given function,
x5/5 –cos x + x +c.
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Find the antiderivative for the given function f(x) = x5 +x +11?

For solving Antiderivative we need to follow the steps shown below:
Step 1: In the first step we write the given function.
f (x)= x5 + x +11,

Step 2: Now we integrate the both sides of the function,
∫f(x) dx = ∫(x5 + x +11) dx,

Step 3: In this step we will separate the integral function.
∫(x5 + x +11) dx = ∫x5 dx + ∫x dx + ∫11 dx,

Step 4: After above step we will integrate each function with respect to ‘x’.
∫(x5 + x +11) dx = x6/6 + x2/2 + 11x +c [Here x5 integration is x6/6 and Integration of x is x2/2
and integration of 1 is ‘x’],
(Where ‘c’ is integration constant)
At last we get the antiderivative of given function,
X6/6 + x2/2 + 11x +c




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posted:5/11/2012
language:English
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