Question

`int(((x^2 + 2)a^((x + tan^-1x)))/(x^2 + 1))dx` is equal to  ______ 

Options

• `log(a)a^(x + tan^-1x) + c`

• `((x + tan^-1x))/(loga) + c`

• `a^(x + tan^-1x)/(loga) + c`

• `loga(x + tan^-1`x) + c

Answer 1

`int(((x^2 + 2)a^((x + tan^-1x)))/(x^2 + 1))dx` is equal to `underline(a^(x + tan^-1x)/(loga) + c)`

Explanation:

Consider, I = `int((x^2 + 2)a^((x + tan^-1x)))/(x^2 + 1)dx`

Let x + tan^–1x = t

⇒ `(1 + 1/(1 + x^2))dx` = dt

⇒ `(2 + x^2)/(1 + x^2)dx` = dt

So, I = `int a^t  dt`

= `a^t/loga + c`

= `a^(x + tan^-1x)/(loga) + c`