Question

The value of `sqrt(2) int (sinx  dx)/(sin(x - π/4))` is ______.

Options

• `x - log|cos(x - π/4)| + C`

• `x + log|cos(x - π/4)| + C`

• `x - log|sin(x - π/4)| + C`

• `x + log|sin(x - π/4)| + C`

Answer 1

The value of `sqrt(2) int (sinx  dx)/(sin(x - π/4))` is `underlinebb(x + log|sin(x - π/4)| + C)`.

Explanation:

Let I = `sqrt(2) int sinx/(sin(x -  π/4)) dx`

Put `x - π/4` = t

`\implies` dx = dt

∴ I = `sqrt(2) int (sin(π/4 + t))/sint dt`

= `sqrt(2) int (sin  π/4 cos t + cos  π/4 sin t)/sint dt`

= `sqrt(2) int (1/sqrt(2) cost/sint + 1/sqrt(2))dt`

= `int (cot t + 1) dt`

= `log | sin t| + t + C_1`  

= `x + log|sin(x - π/4)| + C`  ...`[∵ C_1 = π/4 = C]`