Question

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

पर्याय

• `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:

1. Identify the integral: We need to evaluate `int (1 - x)^-3 dx`.

2. Use substitution: Let u = 1 – x.

Differentiating both sides with respect to x gives `(du)/dx = -1`, which means dx = – du.

3. Substitute into the integral: `int (1 - x)^-3 dx`

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

= `- int u^-3 du`

4. Apply the Power Rule for Integration: The rule states that `int u^n du = u^(n + 1)/(n + 1) + c` for n ≠ –1.

Here n = –3, so:

`- int u^-3 du = - (u^(-3 + 1)/(-3 + 1)) + c`

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

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

5. Substitute back: Replace u with (1 – x):

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