Question

Evaluate :

$\int \limits_ {0}^{\pi} {e}^{ | \cos x| }[ \: 3 \cos( \frac{1}{2} \cos x) + 2 \sin \: ( \frac{1}{2} \cos x)] \sin x \: dx$
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Answer 1

Question :

Evaluate :

$\int \limits_ {0}^{\pi} {e}^{ | \cos x| }[ \: 3 \cos( \frac{1}{2} \cos x) + 2 \sin \: ( \frac{1}{2} \cos x)] \sin x \: dx$

Some useful Results used :

$1) \int e {}^{ax} \sin(bx)dx = \frac{e {}^{ax} }{a {}^{2} + b {}^{2} } (a \sin(bx) - b \cos(bx) ) + c$

$2) \int e {}^{ax} \cos(bx)dx = \frac{e {}^{ax} }{a {}^{2} + b {}^{2} } (a \cos(bx) + b \sin(bx) ) + c$

Solution :

let $\frac{1}{2} \cos(x) = t$

Now differinate with respect to x

$\implies - \sin(x)dx = 2dt$

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$\int \limits_ {0}^{\pi} {e}^{ | \cos x| }[ \: 3 \cos( \frac{1}{2} \cos x) + 2 \sin \: ( \frac{1}{2} \cos x)] \sin x \: dx$

$= \bf( \frac{14e}{5} + \frac{14}{5e} )( \sin( \frac{1}{2} ) ) + ( \frac{8e}{5} - \frac{8}{5e})( \cos( \frac{1}{2} ) )$

Refer to the attachment

Answer 2

Answer:

$\large \boxed{ \sf{\dfrac{24}{5}\left[e ( \cos\dfrac{1}{2} + \dfrac{1}{2} \sin\dfrac{1}{2} ) - 1 \right]}}$

Step-by-step explanation:

Let's assume that,

$I = \displaystyle\int \limits_ {0}^{\pi} {e}^{ | \cos x| }[ \: 3 \cos( \dfrac{1}{2} \cos x) + 2 \sin \: ( \dfrac{1}{2} \cos x)] \sin x \: dx$

Solving further, we will get,

$= > I = \displaystyle\int \limits_ {0}^{\pi} {e}^{ | \cos x| } \: 3 \sin x \cos( \frac{1}{2} \cos x) \: dx+ \displaystyle\int \limits_ {0}^{\pi} 2 {e}^{ | \cos x| } \sin x \sin \: ( \frac{1}{2} \cos x) \: dx \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ..........(i)$

Simplifying further, we will get,

$= > I = \displaystyle\int \limits_ {0}^{\pi} {e}^{ | \cos x| } \: 3 \sin x \cos( \frac{1}{2} \cos x) \: dx - \displaystyle\int \limits_ {0}^{\pi} 2 {e}^{ | \cos x| } \sin x \sin \: ( \frac{1}{2} \cos x) \: dx \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ..........(ii)$

Now, adding eqn (i) and (ii), we get,

$= > 2I = 2\displaystyle\int \limits_ {0}^{\pi} {e}^{ | \cos x | } 3 \sin x \cos( \frac{1}{2} \cos x ) \: dx \\ \\ = > I = \displaystyle\int \limits_ {0}^{\pi} {e}^{ | \cos x | } 3 \sin x \cos( \frac{1}{2} \cos x ) \: dx$

Now, let's assume that,

$\cos(x) = t$

Differentiatiating both sides, we get,

$= > - \sin(x) dx = dt$

Therefore, we will get,

$= > I = 3\displaystyle\int \limits_ { - 1}^{1} {e}^{ |t| } \cos( \frac{t}{2} ) \: dt \\ \\ = > I = 3 \times 2\displaystyle\int \limits_ {0}^{1} {e}^{ |t| } \cos( \frac{t}{2} ) \: dt \\ \\ = > I = \displaystyle\int \limits_ {0}^{1} {e}^{t} \cos( \frac{t}{2} ) \: dt$

Hence, further simplifying, we will get,

$\sf{\dfrac{24}{5}\left[e ( \cos\dfrac{1}{2} + \dfrac{1}{2} \sin\dfrac{1}{2} ) - 1 \right]}$

$\bold\red{Note:-}$ Refer to the attachments.