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# Antiderivative 2^x?

TopAntiderivative is the operation opposite to the differentiation operation. In other words we can call it Integration. So to calculate the Antiderivative we need to find the integral of the function for which we have to find the antiderivative. Suppose we have have a function A( x ), so to find its antiderivative integrate the function A ( x ), we get ∫ A ( x ). dx = B ( x ), here B ( x ) is the antiderivative of A ( x ). If we differentiate B ( x ) then we get back our original function that is A ( x ).
Now we will see the process of finding antiderivative of 2x , to find antiderivative of 2x , we will have to find the integral of 2x . The process is shown below-
Here f ( x ) = 2x
∫ f ( x ) . dx = ∫ 2x . dx,
∫ 2x . dx = ∫ ( e ln (2) ) x . dx, as we know e ln (2) = 2,
∫ e ln (2) . dx = ∫ ( e ln (2) x . dx,
Let u = ln ( 2 ) x ,
Therefore
du = ln ( 2 ) dx,
Now substitute above values, we get-
= ∫ e u (du / ln 2 ),
= (1 / ln 2) ∫ e u . du,
= (1 / ln 2 ) e u + c,
Now substitute back value of ' u ', we get
=> ( 1 / ln 2 ) e ln (2)x + c,
=> ( 1 / ln 2 ) ( e ln (2) ) x + c,
=> (1 / ln 2) 2x + c,
This is how we calculate the antiderivative 2x.