Calculate int_0^1(x+1)^2003dx?

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You need to evaluate the given definite integral, hence, you need to use integration by substitution, such that:

`x + 1 = t => dx = dt`

You need to change the limits of integration, such that:

`x = 0 => t = 1`

`x = 1 => t = 2`

Replacing the variable and the limits of integration, yields:

`int_1^2 t^2003 dt = (t^2004)/(2004)|_1^2`

You need to use the fundamnetal theorem of calculus, such that:

`int_1^2 t^2003 dt = 1/2004(2^2004 - 1^2004)`

`int_1^2 t^2003 dt = (2^2004 - 1)/2004`

Hence, evaluating the given definite integral, using integration by substitution, yields `int_0^1(x+1)^2003 dx = (2^2004 - 1)/2004` .

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