Question:
if $\int\limits_{a}^{b} \frac{x^{n}}{x^{n} + \left(16 - x\right)^{n}} dx = 6$, then
if $\int\limits_{a}^{b} \frac{x^{n}}{x^{n} + \left(16 - x\right)^{n}} dx = 6$, then
Updated On: Jul 6, 2022
- $a = 4, b = 12, n\in R$
- $a = 2, b = 14, n\in R$
- $a = -4, b = 20, n\in R$
- $a = 2, b = 8, n\in R$
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The Correct Option is B
Solution and Explanation
$\int\limits_{a}^{b} \frac{x^{n}}{x^{n} + \left(16 - x\right)^{n}} dx = 6\quad....\left(i\right)$
Let $a + b = 16$, then by property
$ \int\limits_{a}^{b} \frac{\left(16-x\right)^{n}}{\left(16-x\right)^{n }+ x^{n}} dx = 6\quad....\left(ii\right) $
Adding$ \left(i\right)\,\,$ and $\,\left(ii\right)$, we get
$\int\limits_{a}^{b} 1\cdot dx = 12$
$\Rightarrow b - a = 12 $
Solving $a + b = 16$ and $b- a = 12$,
we get $a = 2, b = 14 $ and $n\in R $
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.