Evaluate: \(\int^{z}_{0}(sin x – cos x) dx \hspace{1mm}
Where\hspace{1mm}letter\hspace{1mm}z = \frac{\pi}{4}. (\pi = pi)\)
- \(\sqrt{2 +1}\)
- \(\sqrt{2 }-1\)
- \(-\sqrt{2 }-1\)
-
\(1-\sqrt{2}\)
The correct answer is: D
Explanation
\[
\int_{0}^{z} (\sin x - \cos x) \, dx\]
where \( z = \frac{\pi}{4} \), we have:
\[\int_{0}^{\frac{\pi}{4}} (\sin x - \cos x) \, dx = \left[-\cos x - \sin x\right]_{0}^{\frac{\pi}{4}}\]
Evaluating at the bounds gives:
\[-\sqrt{2} - (-1) = -\sqrt{2} + 1\]
Thus, the value of the integral is:
\[\int_{0}^{\frac{\pi}{4}} (\sin x - \cos x) \, dx = 1 - \sqrt{2}\]