$\int_0^{π/4}\sin x\, d(x-[x])$ is eq

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Definite Integration

Question:

$\int_0^{π/4}\sin x\, d(x-[x])$ is equal to (where [*] denotes the greatest integer function)

Options:

1/2

$1-\frac{1}{\sqrt{2}}$

none of these

Correct Answer:

$1-\frac{1}{\sqrt{2}}$

Explanation:

$∵ 0 ≤ x < π/4$ $∴ [x] = 0$

Then, $\int_0^{π/4}\sin x\,d(x- [x])=\int_0^{π/4}\sin x\,dx$

$=-\{\cos x\}_0^{π/4}=-(\frac{1}{\sqrt{2}}-1)=1-\frac{1}{\sqrt{2}}$