What is ₁³ ≤ft| 1 - {x}^4 | {dx} equal to?

Correct Answer is : 232/5232/5232/5

Correct Answer is : 232/5232/5232/5

Test Index

UPSC NDA 2 2016 Mathematics Solved Paper

Question : 100 of 120

Marks: +1, -0

What is

\int_{1}^{3} \left| 1 - \mathrm{x}^4 \right| \mathrm{dx}

equal to?

Solution:

Concept:

The absolute value function splits the integral at the point where the expression inside changes sign.

Explanation:

The critical point of

|1 - x^4|

[1,3]

x = 1

, where

1 - x^4 = 0

For

x \ge 1

1 - x^4 \le 0

, so

|1 - x^4| = -(1 - x^4) = x^4 - 1

Therefore,

I = \int_1^3 |1 - x^4|\,dx = \int_1^3 (x^4 - 1)\,dx

Integrate:

\int (x^4 - 1)\,dx = \frac{x^5}{5} - x

Evaluate from

1

3

\left(\frac{3^5}{5} - 3\right) - \left(\frac{1^5}{5} - 1\right) = \left(\frac{243}{5} - \frac{15}{5}\right) - \left(\frac{1}{5} - \frac{5}{5}\right) = \frac{228}{5} - \left(-\frac{4}{5}\right) = \frac{232}{5}

Answer:

\frac{232}{5}

(Option D).

Go to Question:

Prev Question

Next Question