17. Integrate sin x* e^cosx dx A) e^sinx + C C) -e^cosx + C B) e^cosx + C D) -e^sinx+C​

Question

17. Integrate sin x* e^cosx dx
A) e^sinx + C
C) -e^cosx + C
B) e^cosx + C
D) -e^sinx+C​

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Rylee 1 month 2021-08-12T18:49:19+00:00 1 Answer 0 views 0

Answers ( )

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    2021-08-12T18:50:34+00:00

    We’re asked to evaluate,

    \displaystyle\longrightarrow I=\int\sin x\cdot e^{\cos x}\ dx\quad\quad\dots(1)

    Substitute,

    \displaystyle\longrightarrow u=e^{\cos x}\quad\quad\dots(2)

    \displaystyle\longrightarrow du=d(e^{\cos x})

    \displaystyle\longrightarrow du=e^{\cos x}\cdot-\sin x\ dx

    \displaystyle\longrightarrow -du=e^{\cos x}\cdot\sin x\ dx

    \displaystyle\longrightarrow \sin x\cdot e^{\cos x}\ dx=-du

    Then (1) becomes,

    \displaystyle\longrightarrow I=\int-du

    \displaystyle\longrightarrow I=-\int du

    \displaystyle\longrightarrow I=-u+C

    From (2),

    \displaystyle\longrightarrow\underline{\underline{I=-e^{\cos x}+C}}

    Hence (C) is the answer.

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