Question

`int (1 - x)^-3 dx` = ______.

Options

• `1/2 (1 - x)^-2 + c`

• `1/2 (1 + x)^-2 + c`

• `1/2 (1 - x)^-2 + x/2 + c`

• `1/2 (1 - x)^-2 - x/2 + c`

Answer 1

`int (1 - x)^-3 dx` = `underlinebb(1/2 (1 - x)^-2 + c)`.

Explanation:

`int (1 - x)^-3 dx`

Let u = 1 – x.

Differentiating both sides with respect to x:

`(du)/dx = -1`

⇒ dx = – du.

Substitute into the integral:

`int (1 - x)^-3 dx`

= `int u^-3 (-du)`

= `- int u^-3 du`

Using the power rule for integration:

`int u^n du = u^(n + 1)/(n + 1) + c`

`- [u^(-3 + 1)/(-3 + 1)] + c`

= `- [u^(-2)/-2] + c`

= `1/2 u^-2 + c`

Substitute u = 1 – x:

`1/2 (1 - x)^-2 + c`