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UPSC NDA II 2025 Mathematics Paper

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Question Numbers: 85-86

For the following two (02) items:

Let

u=∫e^{x}cosxdx

and

v=∫e^{x}sinxdx.

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Question : 85 of 120

Marks: +1, -0

What is u + v equal to?

$-\frac{du}{dx}$
$-\frac{dv}{dx}$
$\frac{du}{dx}$
$\frac{dv}{dx}$

Solution:

Concept:

The sum of the given integrals simplifies to the derivative of one of them because

e^x(\cos x + \sin x) = \frac{d}{dx}(e^x \sin x)

.

Explanation:

We have

u = \int e^x \cos x \, dx

and

v = \int e^x \sin x \, dx

.

Then

u+v = \int e^x \cos x \, dx + \int e^x \sin x \, dx = \int e^x (\cos x + \sin x) \, dx

.

Notice that

\frac{d}{dx}(e^x \sin x) = e^x \sin x + e^x \cos x = e^x (\cos x + \sin x)

.

Therefore,

\int e^x (\cos x + \sin x) \, dx = e^x \sin x

.

But

\frac{dv}{dx} = e^x \sin x

. Hence

u+v = \frac{dv}{dx}

.

Answer:

Option D.

\frac{dv}{dx}

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