Problem 19E

Integrals with general bases Evaluate the following integrals.

Solution 19E Step_1 x Exponential function; Exponential functions have the form f(x) = a . Where ‘a’ is the base , and x is the exponent (or power) . If ‘a’ is greater than ‘1’ the function continuously increases in value as x increases. A special property of exponential functions is that the slope of the function also continuously increases as x increases. x Example ; the graph of y = 2 is ; NOTICE: As x increases , y also increases As x increases , the slope of the graph also increases. The curve passes through (0, 1). All exponential curves of the form f(x) = a , passes through (0,1) , if a>0. The curve doesn’t passes through the x -axis .It just gets closer and closer to the x -axis as we take smaller and smaller x values. Let a is exponential function , and it is continuous for all x.So , the integral of x a is ; a dx = a ( x 1 ln(a) Step-3 Now , we have to evaluate Now , from the above step -2 , the integral value of 1 x (10) dx = ((10) ( 1 = ( 99 ) 1 . 10 ln(10) 1 Therefore , (10) dx = ( 99 ) 1 . 10 ln(10) 1