Antiderivative of a x

Antiderivative of a x Antiderivative of a x First we will see what the meaning of antiderivation is? An antiderivative of a function f can be thought as a function F whose derivative is equal to f or we can say that F' = f.

The method used to solve for antiderivatives is known as anti-differentiation and the function opposite to this is called differentiation, differentiation is used to find the derivatives.

We can understand the concept of antiderivation or antiderivatives with the help of examples shown below Suppose we have a function f( x ) = x^2 , so to calculate antiderivative we will integrate this function , Know More About Antiderivative of Fractions

so ∫ f( x ) dx = ∫ x^2 dx = (x^3) / 3 + c , where 'c' is an arbitrary constant. So F ( x ) = (x^3) / 3 is the antiderivative function .

From above example we can conclude that Antidervation is an operation opposite to the derivative operation. Now we will see how to calculate the antiderivative of a^x . ∫ a^x dx = a^x / ln ( b ) + c Proof of above expression: Use ∫ e^u du = e^u + c as e lna = b , ∫ a^x dx = ∫ [ ( e lna )x ] dx = ∫ e(lna)x dx let u =(lnb)x Therefore du = (lna) dx , Substitute = e^u ( du / lna ) = (1 / lna ) ∫ e^u du Solving the integral , = ( 1 / lna ) ( e^u + c ) = ( 1 / lna ) e^u + c2 ( new contant) Again substitute back u = ( ln a ) x , = ( 1 / ln a ) e(ln a)x + c2 = ( 1 / ln a ) ( e(ln a) )x + c2 = ( 1 / ln a ) ax + c2 = a^x / ln a + c2 Hence proved.

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