solve please.........................
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Answered by
7
Question :
Evaluate :
Some useful Results used :
Now differinate with respect to x
Answered by
0
Question :
Evaluate :
\begin{gathered}\begin{gathered}\int \limits_ {0}^{\pi} {e}^{ | \cos x| }[ \: 3 \cos( \frac{1}{2} \cos x) + 2 \sin \: ( \frac{1}{2} \cos x)] \sin x \: dx \\\end{gathered} < /p > < p > \end{gathered}
0
∫
π
e
∣cosx∣
[3cos(
2
1
cosx)+2sin(
2
1
cosx)]sinxdx
</p><p>
Some useful Results used :
1) \int e {}^{ax} \sin(bx)dx = \frac{e {}^{ax} }{a {}^{2} + b {}^{2} } (a \sin(bx) - b \cos(bx) ) + c1)∫e < /p > < p > ax1)∫e
ax
sin(bx)dx=
a
2
+b
2
e
ax
(asin(bx)−bcos(bx))+c1)∫e</p><p>ax
2) \int e {}^{ax} \cos(bx)dx = \frac{e {}^{ax} }{a {}^{2} + b {}^{2} } (a \cos(bx) + b \sin(bx) ) + c2)∫e < /p > < p >2)∫e
ax
cos(bx)dx=
a
2
+b
2
e
ax
(acos(bx)+bsin(bx))+c2)∫e</p><p>
let \frac{1}{2} \cos(x)let
2
1
cos(x)
\huge\fcolorbox{black}{aqua}{Solution:-}
Solution:-
Now differinate with respect to x
\implies - \sin(x)dx = 2dt < /p > < p >⟹−sin(x)dx=2dt</p><p>
\begin{gathered} < /p > < p > \begin{gathered}\int \limits_ {0}^{\pi} {e}^{ | \cos x| }[ \: 3 \cos( \frac{1}{2} \cos x) + 2 \sin \: ( \frac{1}{2} \cos x)] \sin x \: dx \\\end{gathered} \end{gathered}
</p><p>
0
∫
π
e
∣cosx∣
[3cos(
2
1
cosx)+2sin(
2
1
cosx)]sinxdx
= \bf( \frac{14e}{5} + \frac{14}{5e} )( \sin( \frac{1}{2} ) ) + ( \frac{8e}{5} - \frac{8}{5e})( \cos( \frac{1}{2}=(
5
14e
+
5e
14
)(sin(
2
1
))+(
5
8e
−
5e
8
)(cos(
2
1
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