Question

How to calculate this integral?`int_0^1x2^xdx`

Answer 1

The integral equals `2/ln2 - 1/(ln2)^2 ~~ 0.804`

Explanation:

Use integration by parts, which states that `int u dv = uv - int v du`. We let `u = x` and `dv = 2^x`. Then `du = dx` and `v = 2^x/ln2`. For now, let's just consider the indefinite integral `int x2^xdx`.

`int x 2^x dx = x(2^x/ln2) - int 2^x/ln2 * 1 dx`

`int x2^xdx = x(2^x/ln2) - int 2^x/ln2 dx`

`int x2^xdx = (x2^x)/ln2 - 1/ln2 int 2^x dx`

`int x2^x dx = (x2^x)/ln2 - 1/ln2(2^x/ln2)`

`int x2^x dx = (x2^x)/ln2 - 2^x/(ln2)^2`

This has been calculated. We now evaluate using the second fundamental theorem of calculus which states that `int_a^b F(x) dx = f(b) - f(a)`, where `F(x)` is continuous on `[a, b]` and `f'(x) = F(x)`.

`int_0^1 x2^x dx = (1(2^1))/ln2 - 2^1/(ln2)^2 - ((0(2^0))/ln2 - 2^0/(ln2)^2)`

`int_0^1 x2^x dx = 2/ln2 - 2/(ln2)^2 + 1/(ln2)^2`

`int_0^1 x2^x dx = 2/ln2 - 1/(ln2)^2`

A numerical approximation of this would give

`int_0^1 x2^x dx ~~ 0.804`

Hopefully this helps!

Answer 2

`int \ x2^x \ dx = (2ln2 - 1)/ln^2 2`

Explanation:

If you are studying maths, then you should learn the formula for Integration By Parts (IBP), and practice how to use it:

`int \ u(dv)/dx \ dx = uv - int \ v(du)dx \ dx`, or less formally
`" " int \ udv=uv-int \ vdu`

I was taught to remember the less formal rule in word; "The integral of udv equals uv minus the integral of vdu". If you struggle to remember the rule, then it may help to see that it comes a s a direct consequence of integrating the Product Rule for differentiation.

Essentially we would like to identify one function that simplifies when differentiated, and identify one that simplifies when integrated (or is at least is integrable).

So for the integrand `x 2^x`, hopefully you can see that `x` simplifies when differentiated and `2^x` effectively remains unchanged under integration.

Let `{ (u,=x, => (du)/dx=1), ((dv)/dx,=2^x, => v=2^x/ln2 ) :}`

Then plugging into the IBP formula gives us:

`\ \ \ int \ (x)(2^x) \ dx = (x)(2^x/ln2) - int_0^1 \ (2^x/ln2)(1) \ dx`
`:. int_0^1 \ x2^x \ dx = [(x2^x)/ln2]_0^1 - 1/ln2 \ int_0^1 \ 2^x \ dx`
`" " = 1/ln2(1*2^1-0) - [1/ln2*2^x/ln2]_0^1`
`" " = 1/ln2(2) - 1/ln^2 2(2^1-2^0)`
`" " = 2/ln2 - 1/ln^2 2(2-0)`
`" " = 2/ln2 - 1/ln^2 2`
`" " = (2ln2 - 1)/ln^2 2`