NDA 2 2025 Maths, question 85

Topic: Integral calculus

Passage · Q85 to Q86

Let $u=\int e^{x}\cos x dx$ and $v=\int e^{x}\sin x dx$ .

Q85

What is $u+v$ equal to?

Solution.

Answer: (d) $\frac{dv}{dx}$

$u = \frac{e^{x}(\sin x + \cos x)}{2}$ and $v = \frac{e^{x}(\sin x - \cos x)}{2}$ .
Adding, $u + v = e^{x}\sin x$ .
And $\frac{dv}{dx} = e^{x}\sin x$ , so $u + v = \frac{dv}{dx}$ .

Trap: Each integral carries a constant of integration; the equality holds when the constants are left out, as is standard.