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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Definite Integration

NDA 2016 · Mathematics · Definite Integration

What is ∫_1^3|1-x^4| dx equal to?

What is ∫13∣1−x4∣dx\int_1^3\left|1-\mathrm{x}^4\right| \mathrm{dx} equal to?

Options

  1. A−232/5-232 / 5
  2. B−116/5-116 / 5
  3. C116/5116 / 5
  4. D232/5232 / 5

Correct answer

D. 232/5232 / 5

Step-by-step solution

I=∫13∣1−x4∣dxI=∫13−(1−x4)dx(∵∣x∣={x,x≥0−x,x<0)I=∫13(x4−1)dx⇒I=[x55−x]13I=(355−3)−(155−1)⇒I=2325\begin{aligned} & I=\int_1^3\left|1-x^4\right| d x \\ & I=\int_1^3-\left(1-x^4\right) d x \quad\left(\because|x|=\left\{\begin{array}{cc}x, & x \geq 0 \\ -x, & x < 0\end{array}\right)\right. \\ & I=\int_1^3\left(x^4-1\right) d x \Rightarrow I=\left[\frac{x^5}{5}-x\right]_1^3 \\ & I=\left(\frac{3^5}{5}-3\right)-\left(\frac{1^5}{5}-1\right) \Rightarrow I=\frac{232}{5}\end{aligned}
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