Question

integration of 2^log x base 4 dx

Answer 1

I hope that it will help you.

Answer 2

Given:

An expression in the exponential-logarithmic form$2^{log_{4}x }$.

To Find:

The integral of the given expression.

Solution:

The given expression can be solved using the concepts of integration, and logarithms.

1. The given expression is$2^{log_{4}x }$.

2. The given expression can also be written as,

=>$2^{log_{2^2}x }$, (4 can be written as 2²)

=>$2^{\frac{1}{2}log_{2}x }$, ( $log_{a^b}x = \frac{1}{b}log_{a}x$)

=> $2^{log_{2} x^{1/2} }$, ($alogb = logb^a$)

=>$x^{1/2}$, ($a^{log_{a} x}$ $= x$).

3. Therefore, the value of the expression$2^{log_{4}x }$ is$x^{1/2}$.

4. The integration of x^1/2 is,

=> ∫$x^{1/2}$, ( ∫$x^n =( \frac{x^{n+1} }{n+1})$ )

=>$\frac{2}{3}x^{3/2}$.

Therefore, the value of the integral of$2^{log_{4}x }$ is (2/3) x^3/2.