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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Definite Integration

NDA 2024 · Mathematics · Definite Integration

Let p=∫_a^b f(x) d x and q=∫_a^b|f(x)| d x . If f(x)=e^ -x , then...

Let p=∫abf(x)dxp=\int_a^b f(x) d x and q=∫ab∣f(x)∣dxq=\int_a^b|f(x)| d x. If f(x)=e−xf(x)=e^{-x}, then which one of the following is correct?

Options

  1. Ap=2qp=2 q
  2. Bp=−qp=-q
  3. C4p=q4 p=q
  4. Dp=qp=q

Correct answer

D. p=qp=q

Step-by-step solution

p=∫abf(x)dxp=\int_a^b f(x) d x and q=∫ab∣f(x)∣dxq=\int_a^b|f(x)| d x∵f(x)=e−x is always +ve∀x∈Re−x=∣e−x∣\begin{aligned} \because \quad f(x) & =e^{-x} \text{ is always }+\mathrm{ve} \forall x \in \mathrm{R} \\ e^{-x} & =\left|e^{-x}\right|\end{aligned}∣f(x)∣=∣e−x∣=e−x=f(x)∴∫abf(x)dx=∫ab∣f(x)∣dx|f(x)|=\left|e^{-x}\right|=e^{-x}=f(x)\therefore \quad \int_a^b f(x) d x=\int_a^b|f(x)| d x Hence, p=q\quad p=q
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