Question

The value of `intx^x (1 + logx)dx` is equal to ______.

Options

• `1/2(1 + logx)^2 + c`

• `x^(2x) + c`

• `x^x.logx + c`

• `x^x + c`

Answer 1

The value of `intx^x (1 + logx)dx` is equal to `bbunderline(x^x + c)`.

Explanation:

`I = intx^x (1 + logx)dx`

We recognize that x^x can be rewritten using the exponential function:

x^x = e^xlogx

Taking the derivative,

`d/(dx)(x^x) = x^x (1 + logx)`

Thus, the given integral simplifies directly to:

`I = int d(x^x) = x^x + c`

= `x^x + c`