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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2025 · Mathematics · Indefinite Integration

For the following two (02) items: Let u = ∫ e^x x\, dx and v = ∫...

For the following two (02) items:
Let u=∫excos⁡x dxu = \int e^x \cos x\, dx and v=∫exsin⁡x dxv = \int e^x \sin x\, dx.

What is u+vu + v equal to?

Options

  1. A−dudx-\dfrac{du}{dx}
  2. B−dvdx-\dfrac{dv}{dx}
  3. Cdudx\dfrac{du}{dx}
  4. Ddvdx\dfrac{dv}{dx}

Correct answer

D. dvdx\dfrac{dv}{dx}

Step-by-step solution

Given u=∫excos⁡x dxu = \int e^x \cos x\, dx and v=∫exsin⁡x dxv = \int e^x \sin x\, dx

Adding uu and vv:

u+v=∫ex(cos⁡x+sin⁡x) dxu + v = \int e^x (\cos x + \sin x)\, dx

Using the standard integral ∫ex(f(x)+f′(x)) dx=exf(x)\int e^x (f(x) + f'(x))\, dx = e^x f(x), where f(x)=sin⁡xf(x) = \sin x and f′(x)=cos⁡xf'(x) = \cos x:

u+v=exsin⁡xu + v = e^x \sin x

From the given expression for vv, differentiating with respect to xx:

dvdx=ddx(∫exsin⁡x dx)=exsin⁡x\dfrac{dv}{dx} = \dfrac{d}{dx} \left( \int e^x \sin x\, dx \right) = e^x \sin x

Therefore, u+v=dvdxu + v = \dfrac{dv}{dx}

Answer: dvdx\dfrac{dv}{dx}
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