NIMCET 2021 — Mathematics PYQ

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Question

NIMCET 2021 — Mathematics PYQ

NIMCET | Mathematics | 2021

The integral $\int e^{x} (sinh x + cosh x) d x$ is equal to:

Choose the correct answer:

Explanation

Step 1: Use the definitions of $\sinh x$ and $\cosh x$ .

The hyperbolic sine and cosine functions are defined as:

\sinh x = \frac{e^{x} - e^{-x}}{2}

\cosh x = \frac{e^{x} + e^{-x}}{2}

Step 2: Add the two functions.

\sinh x + \cosh x = \left( \frac{e^{x} - e^{-x}}{2} \right) + \left( \frac{e^{x} + e^{-x}}{2} \right)

\sinh x + \cosh x = \frac{e^{x} - e^{-x} + e^{x} + e^{-x}}{2}

\sinh x + \cosh x = \frac{2e^{x}}{2} = e^{x}

Step 3: Substitute the simplified expression into the integral.

The integral becomes:

I = \int e^{x} (e^{x}) dx

I = \int e^{2x} dx

Step 4: Integrate with respect to $x$ .

Using the rule $\int e^{ax} dx = \frac{e^{ax}}{a} + C$ :

I = \frac{e^{2x}}{2} + C

Alternative Method (Using $e^x[f(x) + f'(x)]$ property):

Let $f(x) = \sinh x$ , then $f'(x) = \cosh x$ .

We know that $\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$ .

I = e^x \sinh x + C

Substituting $\sinh x = \frac{e^x - e^{-x}}{2}$ :

I = e^x \left( \frac{e^x - e^{-x}}{2} \right) + C = \frac{e^{2x} - 1}{2} + C

Since $\frac{-1}{2}$ is a constant, it can be absorbed into the constant of integration $C$ :

I = \frac{e^{2x}}{2} + C

Final Answer:

\int e^{x}(\sinh x + \cosh x)dx = \frac{e^{2x}}{2} + C

Choose the correct answer:

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