Question

Compute \(\int_2^3 \frac {cos⁡x-sin⁡x}{4}\)dx.

(a) \(\frac {1}{4}\) (sin 2 + cos 3 – sin 3 – cos 2)

(b) \(\frac {1}{4}\) (sin 3 – cos 3 – sin 2 – cos 2)

(c) \(\frac {1}{4}\) (sin 3 + cos 3 – sin 2 – cos 2)

(d) \(\frac {1}{4}\) (sin 3 + cos 3 + sin 2 – cos 2)

I had been asked this question in final exam.

My enquiry is from Definite Integral in division Integrals of Mathematics – Class 12

Answer 1

Right option is (c) \(\frac {1}{4}\) (sin 3 + cos 3 – sin 2 – cos 2)

To elaborate: \(\int_2^3 \frac {cos⁡x-sin⁡x}{4}\)dx = \(\frac {1}{4}\) [sin x – (- cos x)]^32

= \(\frac {1}{4}\) (sin x + cos x)^32

= \(\frac {1}{4}\) (sin 3 + cos 3) – \(\frac {1}{4}\) (sin 2 + cos 2)

= \(\frac {1}{4}\) (sin 3 + cos 3 – sin 2 – cos 2)