NIMCET 2021 Question Paper with solutions

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Question : 35 of 120

Marks: +1, -0

\∫e^{x}(\sinh x+\cosh x) d x

$e^{x} \sec h x+C$
$e^{x} \cos h x+C$
$\sin h 2 x+C$
$\cosh 2 x+C$

Solution:

\∫e^{x}(\sinh x+\cosh x) d x

Let

\sin h x=f(x) {e^{x}-e^{-x}}/{2}

\& \ \cos h x=g(x)={e^{x}+e^{-x}}/{2}

Now,

\∫e^{x}[f(x)+g(x)] d x

\⇒\∫e^{x f}(x) d x+\∫e^{x} g(x) d x

\⇒\∫e^{x} f(x) d x+e^{x} g(x)-\∫e^{x} g'(x)

........(1)

∵g(x)={e^{x}+e^{-x}}/{2}

g'(x)=f(x) .

....(2)
By eq. (1)
We have,

\∫e^{x }f(x) d x+e^{x} g(x)-\∫e^{x} g'(x) d x

By using equation (2)

\⇒\∫e^{x} f(x) d x+e^{x} g(x)-\∫e^{x} f(x) d x+C

\⇒e^{x} g(x)+C

\⇒e^{x} \cos h x+C

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