Question

1.Given ∫ 2^x dx = f(x) + C, then f(x) ​

Answer 1

Given:

 $\int 2^xdx=f(x)+c$

To Find:

f(x)?

Step-by-step explanation:

• We have the equation

               $\int 2^xdx=f(x)+c$

• with the help integeration formula we will solve this which is $\int a^xdx=\frac{a^x}{log a} +C$

                       where C is constant.

• Now using this formula in given equation we get

                      $\int \frac{2^x}{log2}+C =f(x)+C$

Thus,$f(x)= \frac{2^x}{log2}$

Answer 2

$\displaystyle \sf f(x)= \frac{2^x}{log2}$

Given :

$\displaystyle \sf \int 2^xdx = f(x) +C$

To find :

The function f(x)

Solution :

Step 1 of 2 :

Write down the given equation

Here the given equation is

$\displaystyle \sf \int 2^xdx = f(x) +C \: \: \: \: - - - - (1)$

Step 2 of 2 :

Find the function f(x)

We are aware of the formula that

$\boxed{ \: \: \displaystyle \sf \int a^xdx=\frac{a^x}{log a} +C \: \: }$

Taking a = 2 we get

$\displaystyle \sf \int 2^xdx=\frac{2^x}{log 2} +C$

Comparing with the Equation 1 we get

$\displaystyle \sf f(x)= \frac{2^x}{log2}$

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