`int_0^1 (5x - 5^x) dx` Evaluate the integral

Expert Answers
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You need to evaluate the integral, hence, you need to use the fundamental theorem of calculus, such that:

`int_a^b f(x)dx = F(b) - F(a)`

`int_0^1(5x - 5^x)dx = int_0^1(5x)dx - int_0^1 5^x dx`

Evaluating each definite integral, yields:

`int_0^1(5x)dx = 5x^2/2|_0^1 = (5/2)(1^2 - 0^2) = 5/2`

`int_0^1 5^x dx = (5^x)/(ln 5)|_0^1 = (5^1)/(ln 5) - (5^0)/(ln 5)`

`int_0^1 5^x dx = 1/(ln 5)(5 - 1) = 4/(ln 5)`

Hence, evaluating the definite integral, using the fundamental theorem of calculus yields `int_0^1(5x - 5^x)dx = 5/2 - 4/(ln 5).`