Let \(f:R→R\) be defined by f(x)={\(x \text{ if } x≤1 -x+2 \text{ if } x>1\)}.Then \(∫_0^2f(x)dx\)=?
Let \(f:R→R\) be defined by f(x)={\(x \text{ if } x≤1 -x+2 \text{ if } x>1\)}.Then \(∫_0^2f(x)dx\)=?
\(=2(tan^{-1}e-\dfrac{\pi}{4})\)
\(=(tan^{-1}e+\dfrac{\pi}{2})\)
\(=4(tan^{-1}e+\dfrac{\pi}{4})\)
\(=(tan^{-1}e-\dfrac{\pi}{2})\)
\(=(tan^{-1}e-\dfrac{\pi}{4})\)
The Correct Option is A
Solution and Explanation
Given that:
\(∫_0^1 \dfrac{2e^x}{1+e^{2x}} dx\)
Take, \(u = e^x \)
\(du=e^xdx\)
Then corresponding limits will be \(1,e\)
\(\therefore\) \(2∫_0^1 \dfrac{t}{1+t^2} dx\)
=\(2[tan^{-1}t]_1^e\)
\(=2(tan^{-1}e-tan^{-1}(1))\)
\(=2(tan^{-1}e-\dfrac{\pi}{4})\) (_Ans)
Top Questions on Relations and Functions
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
