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NIMCET 2021 Question Paper with solutions

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Question : 35 of 120

Marks: +1, -0

\int e^{x}(\sinh x+\cosh x) dx

$e^{x} \operatorname{sech} x + C$
$e^{x} \cosh x + C$
$\sinh 2x + C$
$\cosh 2x + C$

Solution:

\int e^{x}(\sinh x+\cosh x) dx

Let

\sinh x = f(x) \frac{e^{x}-e^{-x}}{2}

\& \cosh x = g(x) = \frac{e^{x}+e^{-x}}{2}

Now,

\int e^{x}[f(x)+g(x)] dx

\Rightarrow \int e^{x f}(x) dx + \int e^{x} g(x) dx

\Rightarrow \int e^{x} f(x) dx + e^{x} g(x) - \int e^{x} g'(x)

........(1)

\because g(x) = \frac{e^{x}+e^{-x}}{2}

g'(x) = f(x) .

....(2)

By eq. (1)

We have,

\int e^{x} f(x) dx + e^{x} g(x) - \int e^{x} g'(x) dx

By using equation (2)

\Rightarrow \int e^{x} f(x) dx + e^{x} g(x) - \int e^{x} f(x) dx + C

\Rightarrow e^{x} g(x) + C

\Rightarrow e^{x} \cosh x + C

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