Question

If I = $\int_{0}^{\pi}{e^{|\cos x|}\left\{ 2\sin\left( \frac{1}{2}\cos x \right) + 3\cos\left( \frac{1}{2}\cos x \right) \right\}}$ sin dx, then I equals

• A. 7$\sqrt{e}$cos (1/2)
• B. 7$\sqrt{e}$ [cos (1/2) – sin (1/2)]
• C. 0
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Put $\frac{1}{2}$cos x = t, so that –sin x dx = 2dt and
I = $\int_{1/2}^{- 1/2}e^{|t|}$ (2 sin t + 3 cost) (–2) dt

As e^|t| sin t is an odd function, and e^|t| cos t is an even function,

I = 6 $\int_{0}^{1/2}{e^{t}\cos tdt}$= 6e^t cos t$\rbrack_{0}^{1/2}$+ 6$\int_{0}^{1/2}{e^{t}\sin tdt}$

I = 6 $\left\lbrack \sqrt{e}\cos\left( \frac{1}{2} \right) - 1 \right\rbrack$+ 6e^t sin t$\rbrack_{0}^{1/2}$– 6 $\int_{0}^{1/2}{e^{t}\cos tdt}$

Ž 7I = 6 $\sqrt { \mathrm { e } }$ $\left( \cos\left( \frac{1}{2} \right) + \sin\left( \frac{1}{2} \right) - 1 \right)$