IGNOU MTE 9 2026 - Verify the second mean value theorem for the function &

Question

Verify the second mean value theorem for the function $f(x) = x$ and $g(x) = \cos x$ in the interval $\left[ 0, \frac{\pi}{2} \right]$ .

24 Mar 2026

Answer :

Word Count : 309

The second mean value theorem for integrals states: if (f(x)) is monotonic on ([a,b]) and (g(x)) is integrable, then there exists a (c \in [a,b]) such that [ \int_a^b f(x) g(x) , dx = f(a) \int_a^c g(x) , dx + f(b) \int_c^b g(x) , dx. ] Here, (f(x) = x) (increasing, hence monotonic) and (g(x) = \cos x) on ([0, \pi/2]). First, calculate the left-hand side: [ \int_0^{\pi/2} x \cos x , dx. ] Use integration by parts: let (u = x \Rightarrow du = dx), (dv = \cos x , dx \Rightarrow v = \sin x). Then [ \int x \cos x , dx

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