Question

integrate sinx-cosx from 0 to pi/4

Answer 1

Correct Question :

$\tt\int\limits_{0}^{\frac{\pi}{4}} sin \: x - cos \: x \: dx.$

Answer :

√2 - 1.

Solution :

$\tt \mapsto\int\limits_{0}^{\frac{\pi}{4}} sin \: x - cos \: x \: dx.$

Let Apply Addition method, Write both separately.

$\tt \mapsto\int\limits_{0}^{\frac{\pi}{4}} sin \: x \: dx - \int\limits_{0}^{\frac{\pi}{4}} cos \: x \: dx.$

$\tt \mapsto[- cos \: x]_{0}^{\frac{\pi}{4}} - [sin \: x]_{0}^{\frac{\pi}{4}}$

$\tt \mapsto[- cos \:{\frac{\pi}{4} - ( - cos \: 0)]} - [sin {\frac{\pi}{4} - (sin \: 0)]}$

$\tt\mapsto[ - \frac{1}{ \sqrt{2} } + 1]- [ \frac{1}{{ \sqrt{2} } } - 0 ]$

$\tt \mapsto1 - \dfrac{1}{ \sqrt{2} } - \dfrac{1}{ \sqrt{2} }$

$\tt \mapsto \dfrac{ \sqrt{2} - 1 - 1}{ \sqrt{2} }$

$\tt \mapsto \dfrac{ \sqrt{2} - 2}{ \sqrt{2} }$

$\tt \mapsto \dfrac{ \sqrt{2} - 2}{ \sqrt{2} } \times \dfrac{ \sqrt{2} }{ \sqrt{2} }$

$\tt \mapsto \dfrac{2 - 2\sqrt{2}}{ 2}$

$\tt \mapsto \dfrac{2(1 - \sqrt{2}) }{2}$

$\tt \mapsto \dfrac{ \cancel{2}(1 - \sqrt{2}) }{ \cancel2}$

$\tt\mapsto 1 - \sqrt{2} .$

OR, Taking negative sign Common, We get.

$\tt \mapsto \sqrt{2} - 1.$

Therefore, the value of given integration is √2 - 1.